Ruppeiner Geometry and Thermodynamic Phase Transition of the Black Hole in Massive Gravity

Eur. Phys. J. C (2021) 81:626 https://doi.org/10.1140/epjc/s10052-021-09407-y

Regular Article - Theoretical Physics

Ruppeiner geometry and thermodynamic phase transition of the black hole in massive gravity

Bin Wu1,2,3,4,a , Chao Wang1,b, Zhen-Ming Xu1,2,3,4,c, Wen-Li Yang1,2,3,4,d 1 School of Physics, Northwest University, Xi’an 710127, China 2 Institute of Modern Physics, Northwest University, Xi’an 710127, China 3 Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi’an 710127, China 4 Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China

Received: 3 March 2020 / Accepted: 4 July 2021 © The Author(s) 2021

Abstract The phase transition and thermodynamic geom- understand black hole systems. Inspired by the Anti-de Sit- etry of a 4-dimensional AdS topological charged black hole ter/conformal field theory (AdS/CFT) correspondence [5–7], in de Rham, Gabadadze and Tolley (dRGT) massive grav- one can relate the Hawking–Page phase transition [8] with the ity have been studied. After introducing a normalized ther- confinement/deconfinement phase transition of gauge field modynamic scalar curvature, it is speculated that its value [9]. Then particular attention has been paid to the phase tran- is related to the interaction between the underlying black sition of black holes in the AdS spacetime. Recently, the hole molecules if the black hole molecules exist. We show introduction of variation of cosmological constant in the that there does exist a crucial parameter given in terms of first law of black hole thermodynamics has been extensively the topology, charge, and massive parameters of the black studied in literatures [10–15]. Such a variation seems to be hole, which characterizes the thermodynamic properties of awkward, as we are considering black hole ensembles with the black hole. It is found that when the parameter is posi- different theories, while the standard thermodynamics is con- tive, the singlet large black hole phase does not exist for suf- cerned with changes of the state of the system in a given the- ficient low temperature and there is a weak repulsive inter- ory. However, there are some considerable reasons why the action dominating for the small black hole which is sim- variation of is feasible. (i) At first, a number of researches ilar to the Reissner–Nordström AdS black hole; when the support for variable , for example, the four-dimensional parameter is negative, an additional phase region describing cosmological constant represents the varying energy density large black holes also implies a dominant repulsive interac- of a 4-form gauge field strength [16,17]. (ii) What’s more, tion. These constitute the distinguishable features of dRGT one may consider ‘more fundamental’ theories, and the gen- massive topological black hole from those of the Reissner– eral gravity theory we considered could be regarded as the Nordström AdS black hole as well as the Van der Waals fluid effective field theory of supergravity theory. The cosmolog- system. ical constant could be regarded as a parameter in a solution of 10-dimensional supergravity, and is no more fundamental to the theory than a black hole mass or any other param- 1 Introduction eter in the metric. The cosmological constant or the other physical constants are not fixed a priori, but arise as vacuum It hints a fascinating set of connections between black hole expectation values and hence can vary, so that we are always physics and thermodynamics since the pioneering works of referring to the same field theory in this sense. (iii) In the Hawking and Bekenstein about the temperature and entropy presence of a cosmological constant, the scaling argument of black holes [1–4]. The thermodynamics theory, which is is no longer valid and the Smarr relation would be incon- widely used in ordinary systems, is now a useful tool for us to sistent with the thermodynamic first law of black hole if the is fixed. Meanwhile, the pressure-volume term, which is necessary to appear in everyday thermodynamics, would be a e-mail: [email protected] b naturally introduced if the negative cosmological constant e-mail: [email protected] is associated with the positive pressure as P =− and c e-mail: [email protected] (corresponding author) 8πG its conjugate quantity is the thermodynamic volume. This is d e-mail: [email protected]

0123456789().: V,-vol 123 626 Page 2 of 15 Eur. Phys. J. C (2021) 81:626 also a practical reason for the Smarr relation to holding in the molecules that make up the black hole. And we speculate case of asymptotically AdS black holes. Hence the thermo- that the empirical observation of the thermodynamic curva- dynamics of the AdS black hole with varying cosmological ture corresponding to the interaction among the constituent constant would be reasonable. molecules of the system also applies to black holes. The introduction of the extended phase space enriches the So the application of the Ruppeiner geometry into black behaviors of the black hole thermodynamics, like that the hole thermodynamics makes it possible to glimpse the under- small/large phase transition of the charged AdS black hole is lying microstructure of black holes from macroscopic ther- found in analogy with the liquid/gas phase transition of Van modynamic quantities. With the help of Ruppeiner geometry, der Waals system [18]. In the line of this approach, many interesting features are observed in other black hole systems efforts have been done to reveal the abundant phase structure [38–57]. Especially, in some situations that the entropy and and critical phenomena of black holes in such an extended thermodynamic volume of these black holes are not inde- phase space [19–30]. This intriguing subject is come to know pendent, such as Schwarzchild-AdS black hole, Reissner– as black hole chemistry, which associates with each black Nordström AdS (RN-AdS) black hole and etc., the heat hole parameter a chemical equivalent in representations of capacity at constant volume of the black hole becomes zero. the first law of black hole thermodynamics [31,32]. A vanishing heat capacity brings about a divergent thermody- Despite the immense conceptual and phenomenological namic curvature. Consequently, some micro information of success of black hole thermodynamics, the description of the associated black hole is missing from the thermodynamic the microstructure of the black hole remains a huge chal- geometry approach. To cure this problem, either we can con- lenge in current researches. In the literature [33], certain sider other suitable thermodynamic characteristic function unknown black hole micromolecules are proposed, which as the fundamental starting point of Ruppenier geometry provides some intuitions and new insights for the underlying [58,59], or introduce a normalized scalar curvature of ther- microstructure of black holes. In this significant approach, modynamic geometry [60]. In this paper, we adopt the latter. the Ruppeiner geometry and the number density of the Novel thermodynamical features of black hole systems are microstates of the black hole are introduced. Originally, Rup- observed with the help of normalized thermodynamic curva- peiner geometry is mainly to use the Hessian matrix structure, which shows the distinguishable behaviors from that of ture to represent the thermodynamic fluctuation theory [34– Van der Waals fluid [61,62]. 36]. A Ruppeiner metric (line element) measuring the dis- In light of these rich phenomenologies, we are in anticipa- tance between two neighboring fluctuation states is defined. tion of things to be more intriguing when the massive gravity The sign of the Ruppeiner scalar curvature may qualitatively model is considered. The first attempt to endow the graviton imply the interaction between the modules of the fluid sys- with mass is the work of Fierz and Pauli in the context of tem. Specifically, a positive (or negative) thermodynamic linear theory [63]. Unfortunately, the predictions of Fierz– scalar curvature maybe indicate a repulsive (or attractive) Pauli linear massive gravity theory in the massless limit do interaction. Moreover, a noninteracting system such as the not coincide with those of general gravity, which is called ideal gas corresponds to the flat Ruppeiner metric. Besides, van DamVeltman–Zakharov discontinuity [64,65]. Further- it is found that the curvature divergent at the critical point of more, the nonlinear level in a generic Fierz–Pauli theory will phase transition [37]. suffer the so-called Boulware–Deser ghost [66]. These seem For black hole systems, because there does not exist a to put an end to the massive gravity theory, until de Rham, complete theory of quantum gravity, although the most likely Gabadadze and Tolley (dRGT) [67] put out a special class of candidate theories – string theory and loop quantum grav- nonlinear massive gravity that is “Boulware–Deser” ghost- ity theory – have achieved good results to some extent, free [68]. It is found that in dRGT massive gravity theory, the the exploration of the microscopic structure of black holes effective cosmological constant can emerge from the massive is bound to some speculative assumptions. Because of the parameters, rather than a fundamental quantity appearing in development of black hole thermodynamics, we maybe spec- the action of the theory. The value of the effective cosmologi- ulate some micro-dynamics of the black hole from its ther- cal constant is determined by the value of the massive param- modynamics. This is in some sense the reverse process of eters in the massive gravity model. In the case of the effective statistical physics, in which the macroscopic quantities are cosmological constant is positive, the massive gravity theory obtained from the microscopic state of the systems. The idea takes the advantage that provides a possible explanation for of this process is reflected in the exploration of the underlying the accelerated expansion of the universe without any dark microscopic behavior of black holes by the Ruppeiner ther- energy. modynamic geometry. Furthermore, it is still unclear about In addition, the dRGT massive gravity theory also admits the constituents of black holes, hence the abstract concept of the black hole solution in the case of a negative effective cos- black hole molecule may be a good choice. Through anal- mological constant. In the literature [69], Vegh developed a ogy, we can imagine that there is an interaction among the nontrivial black hole solution in four dimensional massive 123 Eur. Phys. J. C (2021) 81:626 Page 3 of 15 626 gravity with a negative cosmological constant. He found that of equilibrium thermodynamics provides a possible way to the mass of graviton behaves like a lattice excitation and explore the microscopic structure of black holes. exhibits a Drude peak in the dual field theory within the Consider an equilibrium isolated thermodynamic system framework of gauge/gravity duality. The feature that mas- with total entropy S, and divide it into two subsystems of sive gravity in AdS background introducing the momentum different sizes. One of these two sub-systems is small SB, dissipation in the dual boundary field theory attract great another one is a large subsystem SE and can be regarded as a interests in the application of gauge/gravity duality, such as thermo-bath. We additionally require that SB SE ∼ S.So the study on the temperature dependence of the shear vis- the total entropy of the system describing by two independent cosity to entropy density ratio [70], the holographic plasmon thermodynamic parameters x0 and x1 have the form of relaxation [71]. Later, it was generalized to study the corre- 0 1 0 1 0 1 sponding thermodynamical properties and phase structures S(x , x ) = SB(x , x ) + SE (x , x ). [72,73]. More black hole solutions and their thermodynamical properties were addressed in arbitrary dimensional dRGT For a system in an equilibrium state, the entropy S is at its massive gravity theory with diverse corrections [74–78]. One local maximum value. Expanding the total entropy at the of the most prominent features that the thermodynamical μ = μ vicinity of the local maximum (x x0 ), we have behaviors of dRGT massive gravity have provided is that the so-called small/large black hole phase transition for the ∂ S μ ∂ S μ S = S + B x + E x 0 ∂ μ μ B ∂ μ μ E charged AdS black holes always exist, no matter the hori- x x x x ( = ) ( = ) 0 0 zon topology is spherical k 1 , Ricci flat k 0 or 2 2 1 ∂ S μ ν 1 ∂ S μ ν ( =− ) + B x x + E x x +···, hyperbolic k 1 [79,80]. This is the significant differ- ∂ μ∂ ν μ B B ∂ μ∂ ν μ E E 2 x x x 2 x x x ence that the Van der Waals-like phase transition was usually 0 0 only recovered in a variety of spherical horizon black hole here S is the local maximum at x0 . The entropy of the equi- backgrounds. Subsequently, the abundant thermodynamical 0 μ librium isolated system is conserved under the virtual change behaviors of the black holes in dRGT massive gravity were which indicates that the first derivative of it is zero, so we investigated, such as the reentrant phase transitions, tricritical promptly arrive at point [81–84]. It is also suggested that the phase transition of the dRGT black holes could relate to the Quasinormal mode 2 2 1 ∂ S μ ν 1 ∂ S μ ν [85,86] and the holographic entanglement entropy [87]. S = S − S = B x x + E x x +··· 0 ∂ μ∂ ν μ B B ∂ μ∂ ν μ E E 2 x x x 2 x x x In the literature [88], the authors find that the singulari- 0 0 2 ties of scalar curvatures which are constructed from HPEM 1 ∂ SB μ ν ≈ x x . (1) 2 ∂xμ∂xν μ B B metric and the Gibbs free energy metrics coincide with the x0 critical point of phase transition for the spherical black hole in dRGT massive gravity. Since the entropy SE of the bath as an extensive thermody- Motivated by these interesting results, in this paper, our namical quantity gets the same order as that of the whole aim is to investigate the Ruppeiner geometry with normalized system, so its second derivatives with respect to the intensive μ scalar curvature proposed in [60] and the phase transition of thermodynamical quantities x are much smaller than those topological black holes in dRGT massive gravity. The outline of SB and has been ignored in the second line. of this paper is as follows. In Sect. 2, we briefly review the In Ruppeiner geometry, the entropy is treated as the ther- Ruppeiner geometry, and derive the scalar curvature of black modynamical potential and its fluctuation S is associated holes in the temperature and volume phase space {T, V }.In with the line element l2 with thermodynamical coordinates μ Sect. 3, we study the thermodynamic properties of the topo- {x }[36]. The Ruppeiner line element reads as logical black hole in dRGT massive gravity. The normalized scalar curvature and critical phenomena of these black holes 2 1 R μ ν l = gμνx x , (2) are investigated in Sect. 4. We end the paper with a summary kB and discussion in Sect. 5. Throughout this paper, we adopt the units h¯ = c = kB = G = 1. where kB is the Boltzmann constant and the Ruppeiner metric R gμν is

∂2 2 Ruppeiner thermodynamic geometry R SB gμν =− . (3) ∂xμ∂xν In this section, we are going to provide a brief introduction to the Ruppeiner thermodynamic geometry. Thermody- Here l2 measures the distance between two neighboring R namic geometry which originates from the fluctuation theory fluctuation states. Thus thermodynamic metric gμν has a 123 626 Page 4 of 15 Eur. Phys. J. C (2021) 81:626 large potential to shine some lights on the information about the metric gμν. The mass of the graviton is related to the microstructure of systems. parameter m, ci ’s are dimensionless constants, and Ui ’s are μ ( + )×( + ) Now we set the thermodynamical coordinates x as tem- symmetric polynomials√ constructed from the n 2 n 2 μ μα perature T and volume V , and take the Helmholtz free energy matrix K ν = g fνα, they take the forms of as the thermodynamical potential. The line element l2 is given by [60] U1 =[K], 2 2 C (∂ P) U2 =[K] −[K ], l2 = V T 2 − V T V 2, (4) 2 3 2 3 T T U3 =[K] − 3[K][K ]+2[K ], 4 2 2 3 2 2 4 where CV = T (∂T S)V is the heat capacity at constant vol- U4 =[K] − 6[K ][K] + 8[K ][K]+3[K ] − 6[K ]. ume. Adopting the convention used in [36], the scalar cur- √ √ vature of the above line element can be worked out directly, μ ν μ The square root in K represents ( K) ν( K) λ = K λ. and it yields The topological black hole solution allows the following form of metric 1 2 R = T (∂ P) (∂ C )(∂ P − T ∂ , P) + (∂ C ) 2 (∂ )2 V T V V T V V V 2CV V P 2 2 2 2 i j + (∂ )2 + ((∂ )(∂2 ) − (∂ )2) ds =−f (r)dt + g(r)dr + r γijdx dx , (7) CV V P T V CV V P T T,V P + (∂ )( (∂ ) − (∂2 )) . 2T V P T T,T,V P V CV (5) where γij is the metric of a n-dimensional hypersurface with constant scalar curvature n(n − 1)k. In general, the value of It is believed that the sign of the scalar curvature encodes k can be 1, 0or−1 corresponding to spherical, planar and the effective interaction between two microscopic molecules hyperbolic topology, respectively. With a special choice of > < of a given system, i.e., R 0 and R 0 are associated reference metric in [69] with repulsive and attractive interaction, respectively. While R = 0 implies that the repulsive and attractive interaction = ( , , 2γ ), are in balance [36]. Moreover, it is speculated that the diver- fμν diag 0 0 c0 ij (8) gence of the scalar curvature occurs at the critical point of phase transition. Although, because of the absence of a hith- Ui ’s can be obtained as erto underlying theory of quantum gravity, there has been a controversial discussion on whether the characterizations U = nc /r, of thermodynamic scalar curvature can be directly general- 1 0 U = ( − ) 2/ 2, ized to the black hole thermodynamics or not [89]. But the 2 n n 1 c0 r well-established black hole thermodynamics makes the Rup- U = ( − )( − ) 3/ 3, 3 n n 1 n 2 c0 r peiner thermodynamic geometry be plausible to phenomeno- U = n(n − 1)(n − 2)(n − 3)c4/r 4. (9) logically or qualitatively provide the information about inter- 4 0 actions of black holes. In the four-dimensional spacetime (n = 2), it is easy to verify that U3 = U4 = 0, which prevents c3 and c4 from appearing 3 Thermodynamics of dRGT massive gravity in the spacetime metric. It indicates that these two parameters have no effect on the phase structure of the four-dimensional In this section, we would like to briefly review the topological topological black hole in massive gravity. Then the metric dRGT black hole, and discuss the thermodynamic properties function has the form of [72] of it in extended phase space. The action describing a (n + 2)-dimensional charged topology AdS black hole in dRGT 2 2 2 1 r m0 q c0c1m massive gravity [69]is f (r) = = k − − + + r +c2c m2. g(r) 3 r 4r 2 2 0 2 √ 4 (10) 1 n+2 1 μν 2 I = d x −g R − 2 − Fμν F + m ci Ui (g, f ) , 16πG 4 i=1 (6) Here the parameters m0 and q relates to the mass and charge amount of the black hole in which is the cosmology constant, Fμν is the electromagnetic field tensor defined as Fμν = ∂μ Aν −∂ν Aμ with vector ω ω M = 2 m , Q = 2 q. (11) potential Aμ. Moreover, f is the reference metric coupled to 8π 0 16π 123 Eur. Phys. J. C (2021) 81:626 Page 5 of 15 626

1 in which ω2 is the volume of the two-dimensional hyper- The critical point of the black hole phase transition solved surface. Without loss of generality, we will set c0 = 1 and from Eq. (15) contains the absolute value of charge |q|, which m = 1 in our following discussions, but leave c1 and c2 as implies that the result we obtained is symmetric to the sign of free parameters. In the extended phase space, the negative the charge. For convenience, we use the symbol q to replace cosmological constant is treated as the pressure of the black |q| in the following discussion. It is worth noticing from hole P =−/8π, which leads to an interpretation of the Eq. (16) that usually the critical phenomenon of black hole black hole mass as enthalpy. In terms of the definition of pres- only recovered in a variety of spherical topology (k = 1) in sure P and the radius of the horizon r0, the thermodynamics previous works. However, from the above critical quantities, quantities of the black hole, i.e., mass M, temperature T , and we find that all values of k (0, ±1) have the possibility to electromagnetic potential can be obtained as in [73] admit Van der Waals- like behaviors provided k + c2 > 0. This is the significant feature of dRGT massive gravity, that ω r q2 8π Pr2 c r a non-zero parameter m related to the gravity mass greatly M = 2 0 k + c + + 0 + 1 0 π 2 2 enriches the thermodynamics behaviors of topological black 8 4r0 3 2 holes. k + c q2 c T = 2r P + 2 − + 1 , One can easily verify that in the absence of massive param- 0 πr π 3 π 4 0 16 r0 4 eters c1 and c2, i.e., c1 = 0 = c2, the critical coefficient ω q = 2 2, = . defined as the ratio of critical temperature Tc to the multi- S r0 (12) 4 r0 plier of critical pressure and volume Pcvc equals to We can also easily get the thermodynamic volume conjugated Tc = 8, to the pressure [18] (17) Pcvc 3

∂ M 4π π V = = r 3 = v3. which coincides with the case of the normal Van der Waals ∂ 0 (13) P S,Q 3 6 fluid. However, in the presence of massive parameters c1 and c2, the critical coefficient becomes v = where a specific volume [18,19] is defined as 2r0.Itis √ notable that the “specific volume” is not the normal definition Tc = 8 + 6c1q , of specific volume in normal thermodynamic (volume per 3/2 (18) Pcvc 3 (k + c2) unit mass). In what following, we will see that the definition of the specific volume v = 2r0 for the black hole is a natural which receives a correction combined by the parameters choice comparing to the equation of state of the Van der c1, q, k, c2. In the next subsection, we focus on the relation Waals fluid. between the Rupperiner geometry with normalized scalar From Eqs. (12) and (13), the equation of state of dRGT curvature and the phase transition of topological black holes black holes turns to [73] in dRGT massive gravity.

T c k + c q2 P = − 1 − 2 + , (14) 3.1 Phase transition of charged dRGT black hole v 4πv 2πv2 2πv4 In this subsection, the thermodynamical characteristics of which is analog to that of Van der Waals fluid. The thermody- charged dRGT black hole with k + c > 0 which undergoes namics theory tells us that the critical point of second order 2 phase transition are described, and we further explore how phase transition located at the inflection point of pressure, the metic parameters affect the thermodynamic properties the condition is of black holes. We first introduce the reduced parameters

∂ P ∂2 P deﬁned as = 0, = 0. (15) ∂v ∂v2 T T P T v S P = , T = , v = , S = . v It leads to the following critical quantities [73] Pc Tc c Sc √ 3/2 2 c1 2(k + c2) (k + c2) 6|q| Then the equation of state Eq. (14) of the black hole arrives T = + √ , P = ,v = √ . c π c 2 c 4 3 6π|q| 24π|q| k + c2 at (16) 3 3 8 c q 6 2 c q 6 2 P = T 1 + 1 − 1 3v˜ 16 k + c 16 k + c 1 For a spherical topology with k = 1, the volume of the two- 2 2 dimensional hypersurface is ω2 = 4π;whenk = 0or−1, with the 2 1 ω − + . (19) different compaction methods, 2 take different positive numbers [90]. v˜2 3v˜4 123 626 Page 6 of 15 Eur. Phys. J. C (2021) 81:626

(a) (b)

Fig. 1 a Isotherms curves with x = 0.3, 0, −0.3andT = 0.8 from the superheated small black hole and the supercooled large black hole. bottom to top in reduced (P , v)˜ space. b The isobar curve in reduced And the blue curve indicates the black hole is in an unstable phase. (T , S) plane with P = 0.7andx = 0. Black solid curves describe The black dashed line implied by equal area law is the phase transition the small black holes and large black holes. The red curves represent temperature

A small/large black hole phase transition occurs when T < tal dashed line determined by the Maxwell equal area law is

1, this ﬁrst order phase transition can be identiﬁed from the phase transition temperature T0, which indicates the area

Maxwell’s equal area law [26]. It will more suitable for us to under the T (S) curve from S1 to S2 is the same as the area verify the equal area law in (T , S) phase space. After consid- enclosed by the black dashed line and coordinate axis. Thus ering the relation of S and v, the equation of state described we get the following conditions by the reduced parameters turns to S2 3 3 2 2 T0(S2 − S1) = T (S)dS, T0 = T (S1) = T (S2), (21) c1q 6 c1q 6 T 1 + − S1 16 k + c2 16 k + c2

3 1 1 = √ + − . from which we get the solutions S1, S2, and T0, read as P S 1 (20) 4 S 2 6S √ √ − ∓ − Here we notice that there is a combination of parameters 3 P 3 3 P S , = √ , (22) , , , 1 2 c1 q k c2, which takes a crucial role in the equation of state. 2P ⎡ ⎤ √ So it is necessary for us to introduce a combined parameter 3 3 P 3 − P 3 ⎣ c1q 6 2 ⎦ c1q 6 2 c q 6 2 T 1 + − = . x = 1 and investigate its effects on the phase 0 16 k + c 16 k + c 2 16 k+c2 2 2 structure of the massive black hole. (23) With Eq. (19), the isothermal curves T = 0.8 are plotted in Fig. 1a in reduced (P , v)˜ phase space from bottom to top The two particular points (T , S ) and (T , S ) in Fig. 1b with x = 0.3, 0 and −0.3. As shown in the ﬁgure, there 0 1 0 2 describe respectively the saturated small black hole and are two extreme points on the isotherms, and the curves are the saturated large black hole. The Eq. (23)isservedas divided into three parts. It can be seen that as x decreases, the small/large black hole coexistence curve, which can be the two extreme points keep approaching.With Eq. (20), we inversed to construct an isobar curve in reduced (T , S) phase space with P = 0.7 and x = 0 for example. The isobar curve shown ⎧ ⎧ ⎧ ⎡ ⎤ ⎨ ⎨ ⎨ 3 1 c q 6 2 in Fig. 1b is correspondingly divided into three pieces by P = − − T ⎣ + 1 ⎦ 1 2sin⎩ arcsin ⎩1 ⎩ 0 1 + two extreme points. Black solid curves represent small black 3 16 k c2 ⎫ ⎫⎫ holes phase and large black holes phase. The red curves repre- 3 ⎬2⎬⎪⎬⎪ sent the superheated small black hole phase and supercooled − c1q 6 2 . + ⎭ ⎪⎪ (24) large black hole phase. And the blue curve whose slope is 16 k c2 ⎭⎭ negative indicates the black hole is in an unstable phase. The extreme points separating the metastable phase from the Substituting the equation of state Eq. (19) and the relation unstable phase are called spinodal points. The black horizon- V =˜v3 into Eq. (23), the form of the coexistence curve can 123 Eur. Phys. J. C (2021) 81:626 Page 7 of 15 626

(a) (b)

Fig. 2 a V as a function of P is independent of the parameter x. b V as a function of T with x = 0.3, 0 and −0.3 from top to bottom be expressed as phase and the large black hole phase is smaller, the coexis- ⎡ ⎤ tence region in phase space is compressed. What’s more, for 2 2 3 $ positive parameter x = 0.3, we can see from the black curve 5 − 3 6V 3 + 3 + 6V 3 c q 6 2 T = ⎣ + 1 ⎦ V 0 + in Fig. 2b that the reaches inﬁnity when the temperature 2V 16 k c2 T ≤ 3/13, which indicates that the difference between the 3 saturated small black hole and the saturated large black hole 2 + c1q 6 . is huge. It is an intriguing feature of dRGT black hole that we 1 (25) will further discuss in what follows. After performing series 16 k + c2 expansion on V at the critical point, we have √ According to the relation between V and S, we can get the √ 1 51 3 3 5 thermodynamic volumes of the black hole corresponding to V = 3 3(1 − P) 2 + (1 − P) 2 + O(1 − P) 2 , 8 S1 and S2 on the isobar curve ⎡ ⎤ 3 3 √ 3 2 c1q 6 2 1 61 2 c1q 6 2 = + ( − ) 2 + ⎣ + ⎦ ⎛ √ √ ⎞3 V 6 2 1 T 1 8 k + c2 2 16 k + c2 3 − P − 3 − 3 P = ⎝ √ ⎠ , 3 5 V1 (26) ( − ) + O( − ) . 2P 1 T 2 1 T 2 ⎛ ⎞ √ √ 3 Obviously, it can be seen that, like the RN-AdS black hole, − + − ⎝ 3 P 3 3 P ⎠ V2 = √ . (27) the critical exponent of the dRGT massive black hole also 2P takes the universal value of 1/2. Now let’s turn our attention to the unstable phase describe

The change V = V2−V1 is an extremely interesting param- by the solid blue curve in Fig. 1b. The spinodal curve sep- eter in the phase transition which can be treated as an order arating the unstable phase and metastable phase is obtained parameter. In the light of Eqs. (26) and (27), together with Eq. from the condition (∂V P)T = 0, one gets (24), we plot the figures about the change V as functions − 1 − 3 $ − 1 2 3V 3 V c1q 6 of P and T respectively in Fig. 2. Tsp = + 2 16 k + c2 These two figures indicate that V decreases monotonically as P and T increase, and finally becomes zero at the 3 c q 6 2 critical point. The situation of V as a function of T will + 1 . 1 + (28) be more subtle. Figure 2b is the curve of V as a function 16 k c2 = . − . of T with x 0 3, 0 and 0 3 from top to bottom. As Inversing it, we can obtain the expression of the spinodal x decreases the corresponding V decreases. The smaller curves V indicates the difference between the small black hole

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inﬁnite V , leading to the absence of singlet large black hole ) * 3 3 2 − 1 − + c1q 6 2 − c1q 6 2 + π 1 2 cos 3 arccos 1 2 T 1 16 k+c 16 k+c 3 1 2 2 3 = ) * , V1 3 3 (29) 2T 1 + c1q 6 2 − c1q 6 2 16 k+c2 8 k+c2 ) * 3 3 2 + 1 − + c1q 6 2 − c1q 6 2 1 2 cos 3 arccos 1 2 T 1 16 k+c 16 k+c 1 2 2 3 = ) * , V2 3 3 (30) 2T 1 + c1q 6 2 − c1q 6 2 16 k+c2 8 k+c2

< where V1 1 is for the small black hole spinodal curve, while at sufficiently low temperature. It implies that there is a crit- > V2 1 is for the large black hole spinodal curve. ical temperature T = 3/13 with positive combined param- We plot the coexistence curve and spinodal curve in Fig. eter x = 0.3. Below the critical temperature, there are only ( , ) 3 in the reduced P T phase space with the red solid and two phases (singlet small black hole phase and coexistence blue dashed curves, respectively. phase) with different volume. However, above the critical ( = , = ) The black dote is the critical point P 1 T 1 , and temperature, three phases (singlet small black hole phase, the region at the conner marked as orange is the supercritical coexistence phase, and also singlet large black hole phase) black hole phase, in which the small black hole and the large of the system with different volumes exist. So that the differ- black hole are indistinguishable. As shown in the figure, there ence between the small black hole and the large black hole are two spinodal curves separated by the coexistence curve, becomes infinite when the temperature T ≤ 3/13, which is and these three curves meet at the critical point. The purple consistent with the results indicated by Fig. 2b. For x = 0, and yellow regions represent the small black holes phase and the asymptotic value is zero. While for negative x, there is large black holes phase, respectively. The shapes of these an intersection point of the coexistence curve with T = 0 four figures are similar, but there are some interesting things horizontal line at finite volume. So that the area of the region because of the existence of massive parameters: the starting under the coexistence curve, becomes small and small as x point of the coexistence curve changes with x. The coexis- decreases. We speculate that, as the combined parameter x tence curve is the separation between the singlet small black decreases, a smaller change in thermodynamical volume at a = . hole and the singlet large black hole. For positive x 0 3, fixed phase transition temperature would cause the transition = = / the coexistence curve starts at P 0 and T 3 13, which from singlet small black hole phase to singlet large black implies the absence of the singlet large black hole at suffi- hole phase. ciently low temperature. While for negative x, the starting point of the coexistence curve is T = 0 with finite pressure P, which implies the absence of the singlet small black 4 Ruppenier geometry of the black hole hole at sufficiently low pressure. The results restore to RN- = AdS when x 0 that the coexistence curve starts from the In this section, we concentrate our attention on the thermo- origin of the phase space. Figure 4 is the plot of the coexis- dynamic geometry of the system and attempt to reveal some ( , ) tence curve and spinodal curves in the reduced T V phase information about the microstructure of the black holes. As = . − . space with x 0 3, 0 and 0 3. The two metastable phases the situation of RN-AdS black hole, S and V of dRGT black (superheated small black holes and supercooled large black hole are dependent, the heat capacity at constant volume holes) regions and the coexistence phase region are clearly CV = T (∂T S)V vanishes, which results in a singularity of displayed. the thermodynamical metric Eq. (4). Introducing a normal- The red solid and blue dashed curves represent the coexis- ized scalar curvature could be a way for us to explore the tence curve and spinodal curve, respectively. The black holes thermodynamical properties of the black hole [60], which is in different phases are identified with different colors in the defined as figure. We can see that the asymptotic behavior of the coex- + , + , 2 2 2 2 istence curve at large thermodynamical volume V is greatly (∂V P) − T ∂V,T P + 2T (∂V P) ∂V,T,T P = . RN = RCV = . changed by x.Forx 0 3, the coexistence curve asymp- 2 (∂ P)2 = / V totically reaches a finite constant temperature T 3 13 at (31)

After a straightforward calculation, the normalized scalar

123 Eur. Phys. J. C (2021) 81:626 Page 9 of 15 626

(a) (b) (c)

Fig. 3 Phase structure of dRGT black hole in reduced (P , T ) space with x = 0.3, 0 and −0.3. The solid red and blue dashed curves represent the small/large black hole coexistence curve and spinodal curves, respectively

(a) (b) (c)

Fig. 4 Phase structure of dRGT black hole in reduced (T , V ) space yellow, green and pink describe the supercritical black hole, small black with x = 0.3, 0 and −0.3. The solid red curve represents the coexis- hole, large black hole, metastable phases (superheated small black hole tence curve of the small/large black hole, and the blue dashed curve and supercooled large black hole), small/large black hole coexistence represents the spinodal curve. The regions marked as orange, purple, phase, respectively

curvature of dRGT black hole in terms of reduced param- the parameter space is physically allowed. We can check this eters with k + c2 > 0is with sign-changing curves, whose expression are ) * ) * 3 3 3 2 2 c1q 6 2 c1q 6 2 c1q 6 2 V + 3V 3 − 1 −4T V 1 + + V + 3V 3 − 1 8 k+c2 16 k+c2 8 k+c2 RN = ) * . (32) 3 3 2 2 c1q 6 2 c1q 6 2 2 −2T V 1 + + V + 3V 3 − 1 16 k+c2 8 k+c2

In order to better understand the effect of parameter x on the = . black holes, we employ Eq. (32) and set T 0 4 to show the $ 2 3 3 behavior of normalized scalar curvature R as the function of 3V 3 − 1 c1q 6 2 c1q 6 2 N T = + + , sc + 1 + V with x = 0.3 and −0.3inFig.5. We can see that RN has 4V 32 k c2 16 k c2 two negative divergent points determined by the condition (33) ∂V P = 0, which corresponds to the spinodal curve. These two divergent points get close as x decreases. In addition, it is easy to verify from Eq. (32) that these two divergent points and Vsc determined by the equation coincide at V = 1 when T = 1. Figure 5 shows us that the scalar curvature is mostly negative in the region of parameter 3 c˜1q 6 2 2 space V , but also has the possibility to be positive. V + V 3 − = . +˜ sc 3 sc 1 0 (34) The parameter space V for a positive curvature with ﬁxed 8 k c2 temperature T and x can be worked out directly by setting RN > 0, however, it should be careful to examine where Before we proceed with plotting the sign-changing curves, we take an analysis of the roots of the above equation. At 123 626 Page 10 of 15 Eur. Phys. J. C (2021) 81:626

(a) (b)

Fig. 5 The normalized scalar curvature of dRGT black hole as function V with x = 0.3, −0.3, and T = 0.4

first, we inversely solve the equation to the form the singlet large black hole phase disappears for sufficient low temperature for positive x. The interesting things hap- 2 1 − 3V 3 pen in the case when x < 0, which shows us another positive x = , 2V curvature region III since the Eq. (34) has two solutions with x < 0. According to the discussion in the previous section, where the combined parameter x is introduced. Taking the compared with RN-AdS black holes, in addition to small first order derivative of the above equation, we have black holes, the interaction between underlying ‘molecules’ of large black holes also behave as repulsion. What’s more, 2 − dx = V 3 1, as x decreases, the area of the region acting as an attraction dV 2V 2 between the ‘molecules’ also decreases. The critical behaviors of normalized scalar curvature can which indicates that x is monotonically increasing for V > 1 be studied to take the series expansion of R along the satu- N and monotonically decreasing for 0 < V < 1. With the rated small and large black hole curves near the critical point. asymptotic behaviors With the help of Eqs. (24), (26), (27), the normalized scalar curvature along the coexistence curve with different value of x(V = 0) →∞, x(V →∞) → 0, (35) x are plotted in Fig. 7. The blue (red) curve is RN along the coexistence curve we know that for a fixed positive x,theEq.(34) admits one for saturated small (large) black hole. We can see that the solution Vsc, while for a fixed negative x, it admits two solu- values of the scalar curvature of the large black holes are = tions Vsc; when the parameter x 0, we can easily obtain always smaller than those of the small black holes with fixed − 3 the solution V = 3 2 , which is the result of RN-AdS black temperature, and at the critical point, they all go to negative hole. In Fig. 6, we plot the solid red curve describing the infinity. Similar to RN-AdS black hole when the combined coexistence curve, the blue dashed curve representing the parameter x ≥ 0, only for the small black hole that RN spinodal curve, and dashed black curves representing the can be positive at low temperature. Moreover, for positive sign-changing curves of the normalized scalar curvature RN combined parameter x = 0.3, Fig. 7a shows us that RN along with x = 0.3, 0 and −0.3. The region painted in yellow indi- the saturated large black hole curve (solid red line) seems cates that the normalized scalar curvature is positive, while to become negative infinity as the temperature approaching in the white region is negative. As we have already known T → 0. In fact, there is a truncation of the horizontal axis for from Van der Waals fluid system that the equation of state the solid red line, the divergent point of RN for the large black should be inapplicable in the region under the coexistence hole (solid red curve) is located at T = 3/13. The intriguing curve. So that the divergent and sign-changing behaviors of behavior of the solid red curve at low temperature in Fig. scalar curvature in the coexistence phase are excluded from 7a is consistent with the behavior of the solid red curve in the thermodynamical viewpoint. Only attractive interactions Fig. 4a. From the figure of phase structure Fig. 4a, we can between the molecules of Van der Waals fluid are allowed see that the saturated large black hole curve (solid red line) in the hard-core model [91]. However, as depicted in Fig. 6, and spinodal curve (where the thermodynamic curvature RN when the combined parameter x ≥ 0, there is a weak repul- diverges) coincide at the truncated temperature T = 3/13. sive interaction dominating for small black holes which are ThesameasFig.4a, the behaviors of Fig. 7a implies that there in the situation that similar to the RN-AdS black hole, and 123 Eur. Phys. J. C (2021) 81:626 Page 11 of 15 626

(a) (b) (c)

Fig. 6 Characteristic curves of the normalized scalar curvature. The red solid curve is the coexistence curve, and the black and blue dashed curves correspond to the sign-changing curve and divergence curve, respectively. The region painted in yellow represent that RN > 0

(a)(b) (c)

Fig. 7 The behavior of the normalized scalar curvature along the coex- inﬁnity at sufﬁciently low temperature. This behavior is consistent with istence curvature RN with x = 0.3, 0 and −0.3. The solid blue curve that of Fig. 4a, where the saturated black hole curve asymptotically represents a saturated small black hole and the red solid curve represents approaches to the spinodal curve (where the thermodynamic curvature saturated large black hole. The solid red curve in a tends to be negative RN diverges) at T = 3/13

is a critical temperature T = 3/13 with positive x = 0.3. in which SSBH and SLBH represent saturated small black With a fixed temperature below the critical value T < 3/13, holes and saturated large black holes, respectively. The series no singlet large black hole phase exists in the system. When expansions of RN indicate that the critical exponent is 2. The the temperature of the system is larger than the critical one, results can also be checked numerically. Near the critical three phases (the singlet small black hole phase, coexistence point, the normalized scalar curvature is assumed to take the phase, and also singlet large black hole phase) exist with form of [91] different volume. < When x 0, there would be a vanishing point for RN ln|RN |=−a ln(1 − T ) + b. (36) of the saturated large black hole, and it moves to a higher temperature as x decreases. Hence the interaction between We numerically calculate the curvature along the coexistence the ‘molecules’ behaving as repulsion not only dominates curve and list the fitting results in Table 1. We find that the in the small black hole at low temperature but also exists in coefficient a always fluctuates in a small range of around 2. the large black hole with a negative combined parameter x. In the case of the error caused by calculation, we believe that These behaviors are consistent with the analysis from Fig. 6. the critical exponent is 2, which is the same as that of the Finally, we discuss the behavior of curvature towards the RN-AdS black hole. critical point along the coexistence curve. For this purpose, we write RN as a function of T by using equations Eqs. (24), (26), and (27), and the series expansions of it at T = 1 along 5 Summary and discussion the coexistence curves have the following forms

√ In this paper, we studied the phase transition and thermody- + 1 −2 1 x − 3 −1 namic geometry of a four-dimensional charged topological RN (SSBH) =− (1 − T ) + √ (1 − T ) 2 + O(1 − T ) , 8 √2 2 black hole in massive gravity. Unlike the RN-AdS black hole + 1 −2 1 x − 3 −1 that only the spherical topology exhibiting Van der Waals- RN (SLBH) =− (1 − T ) − √ (1 − T ) 2 + O(1 − T ) , 8 2 2 like phase transition, the so-called small/large black hole 123 626 Page 12 of 15 Eur. Phys. J. C (2021) 81:626

Table 1 Coefficients a and b of the fitting formulation along the saturated small black hole (SSBH) and saturated large black hole (SLBH) with x = 0.3, 0 and −0.3 x = 0.3 x = 0 x =−0.3 a (SSBH) 1.98695 2.03720 2.03242 -b (SSBH) 1.91910 2.52126 2.45151 a (SLBH) 2.07444 2.04285 2.03545 -b (SLBH) 2.85544 2.51507 2.41169 phase transition for the charged AdS black holes always exist, subtle. When the combined parameter is non-negative, the no matter the horizon topology is spherical (k = 1), Ricci situation is similar to the RN-AdS black hole, i.e., the cur- flat (k = 0) or hyperbolic (k =−1). So that the phase vature changes its sign at sufficiently low temperature only structure of the topological black hole in massive gravity is for small black holes, that the microstructure interactions significantly different from that of the RN-AdS black hole. could be attractive or repulsive. While for the large black Specificity, the critical behaviors of topological black hole hole, the scalar curvature is always negative, which indicates exhibit provided k + c2 > 0. We find that there does exist that among the microstructures for the large black hole there a combined parameter in terms of the topology, charge, and are only attractive interactions. However, when the combined massive parameter of the massive black hole, which char- parameter is negative, it is intriguing that both for the small acterizes the thermodynamic properties of the black hole. and large black hole there are phase regions admitting pos- The coexistence curve and spinodal curve (the curve sepa- itive curvature (see Figs. 6, 7). Also, we found that there is rate the metastable phase and unstable phase) are obtained no singlet large black hole at sufficiently low temperature analytically and plotted in the reduced phase space (P , T ) with positive x, and a phase transition between the coexis- and (T , V ), respectively. The supercritical phase region, tence phase and singlet large black hole occurs at T = 3/13 metastable phase (superheated small black holes and super- with x = 0.3. These constitute the distinguishable features cooled large black holes) regions, and coexistence phase of dRGT massive topological black hole from that of RN- region are clearly displayed. We considered the effect of the AdS black hole as well as the Van der Waals fluid system. It crucial combined parameter on the phase structure of the would be interesting to extend our discussion to the higher massive black hole. It is found that as the combined parame- dimensional dRGT massive gravity that reentrant phase tran- ter x decreasing, the area of the coexistence phase region in sition, the tricritical point can appear, we leave these issues the phase space decreases. What’s more, we calculated the for future work. change of thermodynamical volume of the saturated small At last, we talk about since in the AdS/CFT correspon- black hole and saturated large black hole, the change with dence the cosmological constant is normally considered to a fixed temperature also decreases as the combined param- be a fixed parameter that does not vary, it is natural to ask eter x decreasing. It indicates the difference between small the interpretation once is treated as a thermodynamic vari- black hole phase and large black hole phase is smaller, thus a able. It is suggested [92,93] that varying cosmological con- smaller change in thermodynamical volume at a fixed phase stant could be viewed as varying the number of colors N of transition temperature would cause the transition from sin- the CFT. Thus the behaviors of the chemical potential of the glet small black hole phase to singlet large black hole phase. boundary field theory can reflect the behaviors of the black In the case of x = 0, the results reduce to that of RN-AdS holes, such as Hawking-Page transition [93] and thermody- black hole. namics phase transition of the bulk gravity [94]. Alterna- Furthermore, we studied the Ruppeiner thermodynamical tively, interpretation suggested that the number of colors N geometry of the topological black hole in massive gravity should be keep fixed, so that one can consider that varying at the phase transition. Since the thermodynamical metric corresponds to varying the volume on which the field theory degenerates because of the non-independence of the entropy resides [95], the ‘holographic Smarr relation’ and also the and thermodynamical volume of the massive black hole, the p − V criticality of boundary CFT [96] is studied. In Ref. normalized scalar curvature was introduced to reveal some [97], the authors argued that varying the cosmological con- information about the microstructure of the black hole. As stant provides an alternative notion of changing an energy we have shown in Fig. 5, the curvature goes to infinity at the scale of the dual theory. So that the associated phase struc- critical points, and divergent points (corresponding to the ture (PV phase space) of the black hole thermodynamics in spinodal points) get close as x decreases. One of the most the extended phase space is conjectured to be dual to an RG- prominent results of the topological massive black hole is flow on the space of field theories. They also investigated the that the sign-changing curves of the Ruppeiner curvature are behaviors of the entanglement entropy and two-point corre-

123 Eur. Phys. J. C (2021) 81:626 Page 13 of 15 626 lation function of the boundary field theory, which might be References good indicators of Van de Waals phase transitions in the bulk black hole thermodynamics. In Ref. [98] the authors also 1. J.D. Bekenstein, Black holes and entropy. Phys. Rev. D 7, 2333– attempted to characterize the shear viscosity to entropy den- 2346 (1973) 2. S.W. Hawking, Particle Creation by Black Holes. Commun. Math. sity ratio of dual QFT through Van der Waals-like behaviors Phys. 43, 199–220 (1975) of AdS black hole. 3. J.M. Bardeen, B. Carter, S.W. Hawking, The four laws of black However, unlike the Hawking-Page transition for the hole mechanics. Commun. Math. Phys. 31, 161–170 (1973) AdS-black hole, which was later understood as a confine- 4. S.W. Hawking, Black holes and thermodynamics. Phys. Rev. D 13, 191–197 (1976) ment/deconfinement phase transition in the boundary CFT 5. J.M. Maldacena, The large N limit of superconformal field theo- via AdS/CFT correspondence, the Van der Waals like phase ries and supergravity. Int. J. Theor. Phys. 38, 1113–1133 (1999). transition (small/large phase transition) behaviors of AdS- arXiv:hep-th/9711200 black hole, is lack of a suitable holographic interpretation 6. S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Gauge theory corre- lators from non-critical string theory. Phys. Lett. B 428, 105–114 at present. It is worth emphasizing that the details of the (1998). arXiv:hep-th/9802109 field theory interpretation are still, to a large extent, open 7. E. Witten, Anti-de Sitter space and holography. Adv. Theor. Math. questions. Furthermore, since the thermodynamic variables Phys. 2, 253–291 (1998). arXiv:hep-th/9802150 of the black hole correspond clearly to the thermodynamic 8. S.W. Hawking, D.N. Page, Thermodynamics of black holes in anti- De Sitter space. Commun. Math. Phys. 87, 577 (1983) variables of the dual CFT, we believe that the Ruppeiner 9. E. Witten, Anti-de Sitter space, thermal phase transition, and con- thermodynamics geometry of AdS-black hole of the Einstein finement in gauge theories. Adv. Theor. Math. Phys. 2, 505–532 gravity theory or the modified gravity theory may also have (1998). arXiv:hep-th/9803131 its counterpart of the dual QFT, which is an intriguing and 10. D. Kastor, S. Ray, J. Traschen, Enthalpy and the mechanics of AdS black holes. Class. Quantum Gravity 26, 195011 (2009). meaningful topic worthing to investigate. arXiv:0904.2765 11. B.P. Dolan, Pressure and volume in the first law of black hole thermodynamics. Class. Quantum Gravity 28, 235017 (2011). arXiv:1106.6260 Note added 12. M. Cvetic, G.W. Gibbons, D. Kubiznak, C.N. Pope, Black hole enthalpy and an entropy inequality for the thermodynamic volume. Some of the results presented in this paper have also been Phys.Rev.D84, 024037 (2011). arXiv:1012.2888 independently obtained in Ref. [99], which appeared in par- 13. B. Mahmoud El-Menoufi, B. Ett, D. Kastor, J. Traschen, Gravi- tational tension and thermodynamics of planar AdS spacetimes. allel on arXiv.org. Class. Quantum Gravity 30, 155003 (2013). arXiv:1302.6980 14. A.Castro,N.Dehmami,G.Giribet,D.Kastor,Ontheuniversality Acknowledgements Bin Wu would like to thank Wei Xu and De- of inner black hole mechanics and higher curvature gravity. JHEP cheng Zou for their helpful discussions. The financial supports 07, 164 (2013). arXiv:1304.1696 from the National Natural Science Foundation of China (Grant Nos. 15. B.P. Dolan, D. Kastor, D. Kubiznak, R.B. Mann, J. Traschen, Ther- 11947208, 12047502), China Postdoctoral Science Foundation (Grant modynamic volumes and isoperimetric inequalities for de Sitter Nos. 2017M623219, 2020M673460), Basic Research Program of Nat- black holes. Phys. Rev. D 87(10), 104017 (2013). arXiv:1301.5926 ural Science of Shaanxi Province (Grant Nos. 2019JQ-081, 2017KCT- 16. J. Brown, C. Teitelboim, Dynamical neutralization of the cosmo- 12), Shaanxi Postdoctoral Science and Double First-Class University logical constant. Phys. Lett. B 195, 177–182 (1987) Construction Project of Northwest University are gratefully acknowl- 17. J. Brown, C. Teitelboim, Neutralization of the cosmological con- edged. The authors thank the anonymous referees for the constructive stant by membrane creation. Nucl. Phys. B 297, 787–836 (1988) suggestions that improve this work greatly. 18. D. Kubiznak, R.B. Mann, P-V criticality of charged AdS black holes. JHEP 07, 033 (2012). arXiv:1205.0559 Data Availability Statement This manuscript has no associated data 19. J.M. Toledo, V.B. Bezerra, Some remarks on the thermodynamics or the data will not be deposited. [Authors’ comment: This is a purely of charged AdS black holes with cloud of strings and quintessence. theoretical study, and all the necessary results and analyses are included Eur. Phys. J. C 79, 110 (2019) in this published article.] 20. S.H. Hendi, M.H. Vahidinia, Extended phase space thermodynamics and P-V criticality of black holes with a nonlinear source. Phys. Open Access This article is licensed under a Creative Commons Attri- Rev. D 88, 084045 (2013). arXiv:1212.6128 bution 4.0 International License, which permits use, sharing, adaptation, 21. S.-W. Wei, Y.-X. Liu, Critical phenomena and thermodynamic distribution and reproduction in any medium or format, as long as you geometry of charged Gauss-Bonnet AdS black holes. Phys. Rev. D give appropriate credit to the original author(s) and the source, pro- 87, 044014 (2013). arXiv:1209.1707 vide a link to the Creative Commons licence, and indicate if changes 22. R.-G. Cai, L.-M. Cao, L. Li, R.-Q. Yang, P-V criticality in the were made. The images or other third party material in this article extended phase space of Gauss-Bonnet black holes in AdS space. are included in the article’s Creative Commons licence, unless indi- JHEP 09, 005 (2013). arXiv:1306.6233 cated otherwise in a credit line to the material. If material is not 23. R. Zhao, H.-H. Zhao, M.-S. Ma, L.-C. Zhang, On the critical phe- included in the article’s Creative Commons licence and your intended nomena and thermodynamics of charged topological dilaton AdS use is not permitted by statutory regulation or exceeds the permit- black holes. Eur. Phys. J. C 73, 2645 (2013). arXiv:1305.3725 ted use, you will need to obtain permission directly from the copy- 24. J.-X. Mo, W.-B. Liu, Ehrenfest scheme for P-V criticality in the right holder. To view a copy of this licence, visit http://creativecomm extended phase space of black holes. Phys. Lett. B 727, 336–339 ons.org/licenses/by/4.0/. (2013) Funded by SCOAP3. 123 626 Page 14 of 15 Eur. Phys. J. C (2021) 81:626

25. N. Altamirano, D. Kubiznak, R.B. Mann, Reentrant phase transi- 49. A. Dehyadegari, A. Sheykhi, A. Montakhab, Critical behavior and tions in rotating anti-de Sitter black holes. Phys. Rev. D 88, 101502 microscopic structure of charged AdS black holes via an alternative (2013). arXiv:1306.5756 phase space. Phys. Lett. B 768, 235–240 (2017). arXiv:1607.05333 26. E. Spallucci, A. Smailagic, Maxwell’s equal area law for charged 50. D. Li, S. Li, L.-Q. Mi, Z.-H. Li, Insight into black hole phase tran- Anti-de Sitter black holes. Phys. Lett. B 723, 436–441 (2013). sition from parametric solutions. Phys. Rev. D 96, 124015 (2017) arXiv:1305.3379 51. M. Kord Zangeneh, A. Dehyadegari, A. Sheykhi, R.B. Mann, 27. H. Xu, W. Xu, L. Zhao, Extended phase space thermodynamics for Microscopic origin of black hole reentrant phase transitions. Phys. third order Lovelock black holes in diverse dimensions. Eur. Phys. Rev. D 97, 084054 (2018). arXiv:1709.04432 J. C 74, 3074 (2014). arXiv:1405.4143 52. Y.Chen, H. Li, S.-J. Zhang, Microscopic explanation for black hole 28. Y.-G. Miao, Z.-M. Xu, Parametric phase transition for a Gauss- phase transitions via Ruppeiner geometry: two competing factors- Bonnet AdS black hole. Phys. Rev. D 98, 084051 (2018). the temperature and repulsive interaction among BH molecules. arXiv:1806.10393 Nucl. Phys. B 948, 114752 (2019). arXiv:1812.11765 29. Y.-G. Miao, Z.-M. Xu, Phase transition and entropy inequality of 53. Y.-G. Miao, Z.-M. Xu, Interaction potential and thermo-correction noncommutative black holes in a new extended phase space. JCAP to the equation of state for thermally stable Schwarzschild Anti- 1703, 046 (2017). arXiv:1604.03229 de Sitter black holes. Sci. China Phys. Mech. Astron. 62, 10412 30. W. Xu, H. Xu, L. Zhao, Gauss-Bonnet coupling constant as a free (2019). arXiv:1804.01743 thermodynamical variable and the associated criticality. Eur. Phys. 54. X.-Y. Guo, H.-F. Li, L.-C. Zhang, R. Zhao, Microstructure and J. C 74, 2970 (2014). arXiv:1311.3053 continuous phase transition of a Reissner-Nordstrom-AdS black 31. A.M. Frassino, R.B. Mann, J.R. Mureika, Lower-dimensional black hole. Phys. Rev. D 100, 064036 (2019). arXiv:1901.04703 hole chemistry. Phys. Rev. D 92, 124069 (2015). arXiv:1509.05481 55. Z.-M. Xu, B. Wu, W.-L. Yang, The fine micro-thermal structures 32. D. Kubiznak, R.B. Mann, M. Teo, Black hole chemistry: thermody- for the Reissner-Nordström black hole. Chin. Phys. C 44, 095106 namics with Lambda. Class. Quantum Gravity 34, 063001 (2017). (2020). arXiv:1910.03378 arXiv:1608.06147 56. A. Ghosh, C. Bhamidipati, Thermodynamic geometry for charged 33. S.-W. Wei, Y.-X. Liu, Insight into the microscopic structure of an Gauss-Bonnet black holes in AdS spacetimes. Phys. Rev. D 101, AdS black hole from a thermodynamical phase transition. Phys. 046005 (2020). arXiv:1911.06280 Rev. Lett. 115, 111302 (2015). arXiv:1502.00386 57. A.N. Kumara, C.L.A. Rizwan, D. Vaid, K.M. Ajith, Critical 34. G. Ruppeiner, Application of Riemannian geometry to the thermo- behavior and microscopic structure of charged AdS black hole dynamics of a simple fluctuating magnetic system. Phys. Rev. A with a global monopole in extended and alternate phase spaces. 24, 488–492 (1981) arXiv:1906.11550 35. G. Ruppeiner, Thermodynamic critical fluctuation theory? Phys. 58. Z.-M. Xu, B. Wu, W.-L. Yang, Ruppeiner thermodynamic geom- Rev. Lett. 50, 287–290 (1983) etry for the Schwarzschild-AdS black hole. Phys. Rev. D 101, 36. G. Ruppeiner, Riemannian geometry in thermodynamic fluctuation 024018 (2020). arXiv:1910.12182 theory. Rev. Mod. Phys. 67, 605–659 (1995) 59. A. Ghosh, C. Bhamidipati, Thermodynamic geometry and inter- 37. G. Ruppeiner, A. Sahay, T. Sarkar, G. Sengupta, Thermodynamic acting microstructures of BTZ black holes. Phys. Rev. D 101(10), geometry, phase transitions, and the widom line. Phys. Rev. E 86, 106007 (2020). arXiv:2001.10510 052103 (2012). arXiv:1106.2270 60. S.-W. Wei, Y.-X. Liu, R.B. Mann, Repulsive interactions and uni- 38. A. Sahay, T. Sarkar, G. Sengupta, Thermodynamic geometry and versal properties of charged anti-de Sitter black hole microstruc- phase transitions in Kerr-Newman-AdS black holes. JHEP 04, 118 tures. Phys. Rev. Lett. 123, 071103 (2019). arXiv:1906.10840 (2010). arXiv:1002.2538 61. S.-W. Wei, Y.-X. Liu, R.B. Mann, Ruppeiner geometry, phase tran- 39. C. Niu, Y.Tian, X.-N. Wu, Critical phenomena and thermodynamic sitions, and the microstructure of charged AdS black holes. Phys. geometry of RN-AdS black holes. Phys. Rev. D 85, 024017 (2012). Rev. D 100, 124033 (2019). arXiv:1909.03887 arXiv:1104.3066 62. S.-W. Wei, Y.-X. Liu, Intriguing microstructures of five- 40. S.A.H. Mansoori, B. Mirza, Correspondence of phase transition dimensional neutral Gauss-Bonnet AdS black hole. Phys. Lett. B points and singularities of thermodynamic geometry of black holes. 803, 135287 (2020). arXiv:1910.04528 Eur. Phys. J. C 74, 2681 (2014). arXiv:1308.1543 63. M. Fierz, W. Pauli, On relativistic wave equations for particles of 41. P.Chaturvedi, A. Das, G. Sengupta, Thermodynamic geometry and arbitrary spin in an electromagnetic field. Proc. R. Soc. Lond. A phase transitions of dyonic charged AdS black holes. Eur. Phys. J. 173, 211–232 (1939) C 77, 110 (2017). arXiv:1412.3880 64. H. van Dam, M.J.G. Veltman, Massive and massless Yang–Mills 42. A. Sahay, R. Jha, Geometry of criticality, supercriticality and and gravitational fields. Nucl. Phys. B 22, 397–411 (1970) Hawking-Page transitions in Gauss-Bonnet-AdS black holes. Phys. 65. V.I. Zakharov, Linearized gravitation theory and the graviton mass. Rev. D 96, 126017 (2017). arXiv:1707.03629 JETP Lett. 12, 312 (1970) 43. P. Chaturvedi, S. Mondal, G. Sengupta, Thermodynamic geometry 66. D.G. Boulware, S. Deser, Can gravitation have a finite range? Phys. of black holes in the canonical ensemble. Phys. Rev. D 98, 086016 Rev. D 6, 3368–3382 (1972) (2018). arXiv:1705.05002 67. C. de Rham, G. Gabadadze, Generalization of the Fierz-Pauli 44. S.-W. Wei, B. Liang, Y.-X. Liu, Critical phenomena and chemical action. Phys. Rev. D 82, 044020 (2010). arXiv:1007.0443 potential of a charged AdS black hole. Phys. Rev. D 96, 124018 68. S.F. Hassan, R.A. Rosen, Resolving the ghost problem in non- (2017). arXiv:1705.08596 linear massive gravity. Phys. Rev. Lett. 108, 041101 (2012). 45. R.-G. Cai, J.-H. Cho, Thermodynamic curvature of the BTZ black arXiv:1106.3344 hole. Phys. Rev. D 60, 067502 (1999). arXiv:hep-th/9803261 69. D. Vegh, Holography without translational symmetry. 46. S.I. Vacaru, P. Stavrinos, D. Gontsa, Anholonomic frames and ther- arXiv:1301.0537 modynamic geometry of 3-D black holes. arXiv:gr-qc/0106069 70. S.A. Hartnoll, D.M. Ramirez, J.E. Santos, Entropy production, 47. J.E. Aman, I. Bengtsson, N. Pidokrajt, Geometry of black viscosity bounds and bumpy black holes. JHEP 03, 170 (2016). hole thermodynamics. Gen. Relativ. Gravit. 35, 1733 (2003). arXiv:1601.02757 arXiv:gr-qc/0304015 71. M. Baggioli, U. Gran, A.J. Alba, M. Tornsö, T. Zingg, Holographic 48. G. Ruppeiner, Thermodynamic curvature and black holes. Springer plasmon relaxation with and without broken translations. JHEP 09, Proc. Phys. 153, 179–203 (2014). arXiv:1309.0901 013 (2019). arXiv:1905.00804 123 Eur. Phys. J. C (2021) 81:626 Page 15 of 15 626

72. R.-G. Cai, Y.-P. Hu, Q.-Y. Pan, Y.-L. Zhang, Thermodynamics of 86. D.-C. Zou, Y. Liu, R.-H. Yue, Behavior of quasinormal modes and black holes in massive gravity. Phys. Rev. D 91, 024032 (2015). Van der Waals-like phase transition of charged AdS black holes in arXiv:1409.2369 massive gravity. Eur. Phys. J. C 77, 365 (2017). arXiv:1702.08118 73. J. Xu, L.-M. Cao, Y.-P. Hu, P-V criticality in the extended phase 87. X.-X. Zeng, H. Zhang, L.-F. Li, Phase transition of holographic space of black holes in massive gravity. Phys. Rev. D 91, 124033 entanglement entropy in massive gravity. Phys. Lett. B 756, 170– (2015). arXiv:1506.03578 179 (2016). arXiv:1511.00383 74. S.H. Hendi, B. Eslam Panah, S. Panahiyan„ Thermodynamical 88. M. Chabab, H. El Moumni, S. Iraoui, K. Masmar, Phase transitions structure of AdS black holes in massive gravity with stringy and geothermodynamics of black holes in dRGT massive gravity. Gauge-gravity corrections. Class. Quantum Gravity 33, 235007 Eur. Phys. J. C 79, 342 (2019). arXiv:1904.03532 (2016).arXiv:1510.00108 89. B.P. Dolan, Intrinsic curvature of thermodynamic potentials for 75. T.Q. Do, Higher dimensional nonlinear massive gravity. Phys. Rev. black holes with critical points. Phys. Rev. D 92, 044013 (2015). D 93, 104003 (2016). arXiv:1602.05672 arXiv:1504.02951 76. P. Li, X.-Z. Li, P. Xi, Black hole solutions in de Rham– 90. Y.C. Ong, Hawking evaporation time scale of topological black Gabadadze–Tolley massive gravity. Phys. Rev. D 93, 064040 holes in anti-de Sitter spacetime. Nucl. Phys. B 903, 387–399 (2016). arXiv:1603.06039 (2016). arXiv:1507.07845 77. B. Mirza, Z. Sherkatghanad, Phase transitions of hairy black holes 91. D. Johnston, Advances in thermodynamics of the van der in massive gravity and thermodynamic behavior of charged AdS waals ﬂuid (Morgan & Claypool Publishers, San Rafael, 2014). black holes in an extended phase space. Phys. Rev. D 90, 084006 arXiv:1402.1205 (2014). arXiv:1409.6839 92. C.V. Johnson, Holographic heat engines. Class. Quantum Gravity 78. S.-L. Ning, W.-B. Liu, Black hole phase transition in massive grav- 31, 205002 (2014). arXiv:1404.5982 ity. Int. J. Theor. Phys. 55, 3251–3259 (2016) 93. B.P. Dolan, Bose condensation and branes. JHEP 10, 179 (2014). 79. S.G. Ghosh, L. Tannukij, P. Wongjun, A class of black holes in arXiv:1406.7267 dRGT massive gravity and their thermodynamical properties. Eur. 94. J.L. Zhang, R.G. Cai, H. Yu, Phase transition and thermodynam- 5 Phys. J. C 76, 119 (2016). arXiv:1506.07119 ical geometry for Schwarzschild AdS black hole in AdS5 × S 80. S.H. Hendi, R.B. Mann, S. Panahiyan, B. Eslam Panah, Van der spacetime. JHEP 02, 143 (2015). arXiv:1409.5305 Waals like behavior of topological AdS black holes in massive 95. A. Karch, B. Robinson, Holographic black hole chemistry. JHEP gravity. Phys. Rev. D 95, 021501 (2017). arXiv:1702.00432 12, 073 (2015). arXiv:1510.02472 81. D.-C. Zou, R. Yue, M. Zhang, Reentrant phase transitions of higher- 96. B.P. Dolan, Pressure and compressibility of conformal ﬁeld theo- dimensional AdS black holes in dRGT massive gravity. Eur. Phys. ries from the AdS/CFT correspondence. Entropy 18, 169 (2016). J. C 77, 256 (2017). arXiv:1612.08056 arXiv:1603.06279 82. M. Zhang, D.-C. Zou, R.-H. Yue, Reentrant phase transitions 97. E. Caceres, P.H. Nguyen, J.F. Pedraza, Holographic entanglement and triple points of topological AdS black holes in Born-Infeld- entropy and the extended phase structure of STU black holes. JHEP massive gravity. Adv. High Energy Phys. 2017, 3819246 (2017). 09, 184 (2015). arXiv:1507.06069 arXiv:1707.04101 98. M. Cadoni, E. Franzin, M. Tuveri, Van der Waals-like behaviour 83. B. Liu, Z.-Y.Yang, R.-H. Yue, Tricritical point and solid/liquid/gas of charged black holes and hysteresis in the dual QFTs. Phys. Lett. phase transition of higher dimensional AdS black hole in massive B 768, 393–396 (2017). arXiv:1702.08341 gravity. Ann. Phys. 412, 168023 (2020). arXiv:1810.07885 99. P.K. Yerra, C. Bhamidipati, Ruppeiner geometry, phase transitions 84. S.R. Chaloshtary, M. Kord Zangeneh, S. Hajkhalili, A. Sheykhi, and microstructures of black holes in massive gravity. Int. J. Mod. S.M. Zebarjad, Thermodynamics and reentrant phase transition Phys. A 35, 22 (2020). arXiv:2006.07775 for logarithmic nonlinear charged black holes in massive gravity. arXiv:1909.12344 85. P. Prasia, V.C. Kuriakose, Quasi normal modes and P-V criticallity for scalar perturbations in a class of dRGT massive gravity around Black Holes. Gen. Relativ. Gravit. 48, 89 (2016). arXiv:1606.01132

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