Reentrant phase transitions and triple points of topological AdS black holes in Born-Infeld-massive gravity

Motivated by recent developments of black hole thermodynamics in de Rham, Gabadadze and Tolley(dRGT) massive gravity, we study the critical behaviors of four-dimensional topological Anti-de Sitter(AdS) black holes in the presence of Born-Infeld nonlinear electrodynamics by treating the cosmological constant as pressure and the corresponding conjugate quantity is interpreted as thermodynamic volume. It shows that besides the Van der Waals-like SBH/LBH phase transitions appears, the so-called reentrant phase transitions (RPTs) are also observed when the coupling coefficients $c_i m^2$ of massive potential and Born-Infeld parameter $b$ satisfy some certain conditions.


I. INTRODUCTION
The Einstein's General Relativity (GR), which describes the graviton is a massless spin-2 particle helped us to understand the dynamics of the Universe [1][2][3]. However, there are some fundamental issues, such as the hierarchy problem in particle physics, the old cosmological constant problem and the origin of late-time acceleration of the Universe still exist in GR [4]. One of the alternating theory of gravity is known as a massive gravity, where mass terms are added into the GR action.
A graviton mass has the advantage to potentially provide a theory of dark energy which could explain the present day acceleration of our Universe [5]. On the other hand, since the quantum theory of massless gravitons is non-renormalizable, a natural question is whether one can build a self-consistent gravity theory if the graviton is massive. The first attempt toward constructing the theory of massive gravity was done by Fierz and Pauli(FP) [6]. With the quadratic order, the FP mass term is the only ghost-free term describing a gravity theory with five degrees of freedom [7]. However, due to the existence of the van Dam-Veltman-Zakharov (vDVZ) discontinuity, this theory cannot recover linearized Einstein gravity in the limit of vanishing graviton mass [8,9].
In particular, Vainshtein [10] proposed that the linear massive gravity can be recovered to GR through the 'Vainshtein Mechanism' at small scales by including non-linear terms in the massive gravity action. Nevertheless, it usually brings various instabilities for the gravitational theories on the non-linear level by adding generic mass terms, since this model suffers from a pathology called a 'Boulware-Deser' (BD) ghost. Later, a new nonlinear massive gravity theory was proposed by de Rham, Gabadadze and Tolley (dRGT) [11][12][13], where the BD ghost [14] was eliminated by introducing higher order interaction terms in the action. Then, Vegh [15,16] constructed a nontrivial black hole solution with a Riccit flat horizon in four-dimensional dRGT massive gravity.
Recent development on the thermodynamics of black holes in extended phase space shows that the cosmological constant can be interpreted as the thermodynamic pressure and treated as a thermodynamic variable in its own right [22,23] in the geometric units G N = = c = 1. Such operation assume that gravitational theories including different values of the cosmological constants fall in the same class, with unified thermodynamic relations. For black hole thermodynamics, the variation of the cosmological constant ensures the consistency between the first law of black hole thermodynamics and the Smarr formula. Moreover, the classical theory of gravity may be an effective theory which follows from a yet unknown fundamental theory, in which all the presently 'physical constants' are actually moduli parameters that can run from place to place in the moduli space of the fundamental theory. Since the fundamental theory is yet unknown, it is more reasonable to consider the extended thermodynamics of gravitational theories involving only a single action, and then all variables will appear in the thermodynamical relations. In the extended phase space, the charged AdS black hole black hole admits a more direct and precise coincidence between the first order small/large black holes (SBH/LBH) phase transition and the Van der Waals liquid-gas phase transition, and both systems share the same critical exponents near the critical point [24]. More discussions in various gravity theories can be found in Refs. . Recently, some investigations for thermodynamics of AdS black holes have been also generalized to the extended phase space in the dRGT massive gravity [47][48][49][50], which show the Van der Waals-like SBH/LBH phase transition in the charged topological AdS black holes. In addition, the deep relation between the dynamical perturbation and the Van der Waals-like SBH/LBH phase transition in the four-dimensional dRGT massive gravity has been also recovered in Ref. [51]. This paper is organized as follows. In Sect. II, we review the thermodynamics of Born-Infeldmassive black holes in the extended phase space. In Sect. III, we study the critical behaviors of four and five dimensional topological AdS black holes in context of P − V criticality and phase diagrams. We end the paper with conclusions and discussions in Sect. IV.

II. THERMODYNAMICS OF d-DIMENSIONAL BORN-INFELD ADS BLACK HOLES
We start with the action of d-dimensional massive gravity in presence of Born-Infeld field [58] where the last four terms are the massive potential associate with graviton mass, c i are the negative constants [21] and f is a fixed rank-2 symmetric tensor. Moreover, U i are symmetric polynomials The square root in K is understood as the matrix square root, ie., ( In the limit b → ∞, it reduces to the standard Maxwell field Consider the metric of d-dimensional spacetime in the following form where h ij dx i dx j is the line element for an Einstein space with constant curvature The constant k characterizes the geometric property of hypersurface, which takes values k = 0 for flat, k = −1 for negative curvature and k = 1 for positive curvature, respectively.
By using the reference metric [21] with a positive constant c 0 , we can obtain Obviously, the terms related with c 3 and c 4 only appear in the black hole solutions for d ≥ 5 and d ≥ 6, respectively [21].
In addition, the electromagnetic field tensor in d-dimensions is given by F tr = q r d 2 , and the metric function f (r) is obtained as [58] where Moreover, m 0 and q are related to the mass M and charge Q of black holes as where Σ k represents the volume of constant curvature hypersurface described by h ij dx i dx j . The electromagnetic potential difference (Φ) between the horizon and infinity reads as Φ = d 2 Then the mass M of the Born-Infeld AdS black hole for massive gravity is given by in terms of the horizon radius r + . Due to existence of the pressure in obtained relation for total mass of the black holes, here the black hole mass M can be considered as the enthalpy H rather than the internal energy of the gravitational system [59].
In addition, the Hawking temperature which is related to the definition of surface gravity on the outer horizon r + can be obtained as and the entropy S of the Born-Infeld AdS black hole reads as It is easy to check that those thermodynamic quantities obey the (extended phase-space) first law of black hole thermodynamics where B, which is a quantity conjugate to b is called the "Born-Infeld vacuum polarization" the thermodynamic volume V [60], which is the corresponding conjugate quantity of P , can be written as The behavior of free energy G is important to determine the thermodynamic phase transition in the canonical ensemble. We can calculate the free energy from the thermodynamic relation

BORN-INFELD-MASSIVE GRAVITY
For further convenience, we denotê HereT denotes the shifted temperature and can be negative according to the value of c 0 c 1 m 2 .
Then, the equation of state of the black hole can be obtained from Eq. (12) To compare with the VdW fluid equation, we can translate the "geometric" equation of state to physical one by identifying the specific volume v of the fluid with the horizon radius of the black Evidently, the specific volume v is proportional to the horizon radius r + , therefore we will just use the horizon radius in the equation of state for the black hole hereafter in this paper.
We know that the critical point occurs when P has an inflection point, where the subscript stands for the quantities at the critical point. The critical shifted temperature is obtained asT and the equation for critical horizon radius r c is given by For later discussions, it is convenient to rescale some quantities in the following way In terms of quantities above, Eqs. (19), (21) and (22) can be written as where x c denotes the critical value of x. For arbitrary parameter d, it is hard to obtain the exact solution of Eq. (26).
In what follows we shall specialize to d = 4 and 5, and then perform a detailed study of the thermodynamics of these black holes.
A. P − V criticality for d = 4 For d = 4, Eq.(26) will reduce to the cubic equation Depending on different values of w 2 , Eq. (27) admits one or more positive real roots for x, which can be also reflected by When |w 2 | ≤ 1 2π , three real roots occur, which are given by Moreover, in order that x c = 1 be positive, we require an additional constraint |y| ≤ 1 √ 2 . Then, we have y 0 > 0 in case of − 1 2π ≤ w 2 ≤ − 1 √ 8π , and y 1 > 0 in the region of − 1 2π ≤ w 2 ≤ 0, while the solution y 2 is always negative. Now by inserting solutions of y 0 and y 1 into Eqs. (24) and (25), we analyze the critical behaviors.
Notice that analytic methods can not be applied in our analysis because of the complexity of the Gibbs free energy and equation of state, we resort to graphical and numerical methods.
1. w 2 ∈ (− 1 √ 8π , 0). As shown in Fig. 1, the p − x diagram displays that the dashed curve represents critical isotherm at t = t c , the dotted and solid curves correspond to t > t c and t < t c , respectively. In the g − t diagram, the solid curve represents p < p c , the dotted curve correspond to p > p c and the dashed curve is for p = p c . We observe standard swallowtail behavior. Moreover, the p − t diagram shows the coexistence line of the first-order phase transition terminating at a critical point. These plots are analogous to typical behavior of the liquid-gas phase transition of the Van der Waals fluid.  3. w 2 ∈ (− 1 2π , −0.132795), there exist two critical points with positive pressure, and the similar RPT also occurs. As shown in Fig. 4, we obtain (t c2 , t τ , t z , t c1 ) ≈ (0.187113, 0.197121, 0.198064, 0.2139999), 0.0116313, 0.018695, 0.0194174, 0.0235228), (31) when taking w 2 = −0.14. With regard to |w 2 | > 1 2π , the solution of Eq. (27) is given by which violates the constraint condition |y| ≤ 1 √ 2 . All in all, when the parameter w 2 satisfies − 1 2π < w 2 < 0, the Van der Waals-like SBH/LBH phase transition appears. In addition, the interesting RPT happens in case of − 1 2π < w 2 < − 1 √ 8π .
B. P − V criticality for d = 5 Then Eq. (26) can be rewritten as Evidently, it is not possible to obtain analytic solution of above equation. To see more closely the phase transition of the Born-Infeld AdS black hole, here we analyze the asymptotic property of the function F (x c ). In addition, the function dF (xc) dxc reads Evidently, Eq. (33) has more than one real roots. For different values of w 2 and w 3 , we will investigate the phase structure and criticality in the extended phase space.
1. w 2 > 0 and w 3 > 0 When x c → +∞, F (x c ) equals to − 2πw 2 27 . Near the origin x = 0, we have dxc is always positive, so there is no a real solution for x + . Therefore, there is no a critical point.  The dashed curve represents critical isotherm at t = t c . The dotted and solid curves correspond to t > t c and t < t c , respectively. Center: the g − t diagram. The solid curve represents p < p c , the dotted curve correspond to p > p c and the dashed curve is for p = p c . We observe standard swallowtail behavior. Right: The p − t diagram, showing the coexistence line of SBH/LBH phase transition terminating at a critical point. These plots are analogous to typical behavior of the liquid-gas phase transition of the VdW'fluid.
3. w 2 > 0 and w 3 < 0 In this case, it is a hard work to discuss the asymptotic property of Eq. (33). Here we resort to graphical and numerical methods, and also find the existence of VdW-like small/large black hole phase transition in the system, see Figure. 6. We plot the pressure p as a function of x for t = 0.48, 0.497, 0.4988, and 0.51 (from bottom to top) in Fig. (7). When p < p c1 , there exists a characteristic swallow tail behavior in the g − t diagram, and a VdW-like SBH/LBH phase transition will occur. Further increasing p such that p c1 < p < p c2 , there appears a new stable IBH branch. For the corresponding Gibbs free energy in Therefore, we observe a triple point characterized by (p τ , t τ ) = (0.01960, 0.11226). Slight above this pressure, the system will emerge a standard SBH/IBH/LBH phase transition with the increase of t . And such phase transition disappears when p c2 is approached.
Further increasing p, the stable IBH branch vanishes in case of p c2 < p < p c3 . And only one stable branch survives when p > p c3 . In the ranges p < p c1 and p c2 < p < p c3 , it displays one characteristic swallow tail behavior in Fig. (8). When p > p c3 , there is no such behavior.

IV. CONCLUSIONS AND DISCUSSIONS
In the extended phase space, we have studied the phase transition and critical behavior of topological AdS black holes in the four and five dimensional Born-Infeld-massive gravity. For d = 4, we found that when the horizon topology is spherical (k = 1), Ricci flat (k = 0) or hyperbolic (k = −1), there always exist the Van der Waals-like SBH/LBH phase transition when the coupling coefficients of massive potential is located in the region − 1 2π < w 2 < 0. In addition, a  monotonic lowering of the temperature yields a large-small-large black hole transition in the region where we refer to the former large state as an intermediate black hole (IBH), which is reminiscent of RPTs. Moreover, this process is also accompanied by a discontinuity in the global minimum of the Gibbs free energy, referred to as a zeroth-order phase transition.
In some range of the parameters, there are three critical points for five-dimensional Born-Infeld AdS black hole. In such range, the Gibbs free energy displays the behavior of two swallow tails. This phenomenon has been never recovered before.
Recent observations of gravitational waves have put an upper bound of 1.2 × 10 −22 eV /c 2 on the graviton's mass [62]. We can find in 4-dimensional case, the interesting RPTs can always appear as long as the parameters q and b take the suitable values with the constant k takes the values ±1.
When the constant k = 0, the role of the graviton's mass is highlighted, the parameters q and b cannot take an acceptable range (means in the framework of the Born-Infeld theory) to make the parameter w 2 ∈ (− 1 2π , − 1 √ 8π ), which means only the VdW-like phase transitions might happen. In the 5-dimensional case, when the constant k = 1, this interesting phenomenon could appear as long as the parameters q and b take the suitable values. There is no three critical points when the constant k takes −1 or 0, because the parameter w 2 is always positive.
Ref. [57] shows that the RPTs only exist in the 4-dimensional Born-Infeld AdS black hole with a spherical horizon , and also gives the proof that there is no reentrant phase transition in the system of higher(≥ 5) dimensional Born-Infeld AdS black hole. Ref. [48] demonstrated that there only exist the Van der Waals like phase transition in the 4-dimensional AdS black hole in massive gravity with Maxwell's electromagnetic field theory. Our results reveal that the nonlinear electromagetic field plays an important role in the phase transition of the 4-dimensional AdS black hole, and the massive gravity could bring richer phase structures and critical behavior (triple critical points) than that of the Born-Infeld term in the 5-dimensional AdS black hole.
Recently, the charged black hole [63], Born-Infeld black hole [64], and black hole in the Maxwell and Yang-Mills fields [65] have been constructed in Gauss-Bonnet-massive gravity. Only Van der Waals like first order SBH/LBH phase transition exist in these models. In addition, this RPT and triple points also occur in the higher-dimensional rotating AdS black holes [66,67], and higherdimensional Gauss-Bonnet AdS black hole [68][69][70]. It would be interesting to extend our discussion to these black holes in Gauss-Bonnet and and 3rd-order Lovelock-massive gravity.