Reentrant phase transitions of higher-dimensional AdS black holes in dRGT massive gravity

Eur. Phys. J. C Reentrant phase transitions of higher-dimensional AdS black holes in dRGT massive gravity De-Cheng Zou 1 Ruihong Yue 1 Ming Zhang 0 0 Faculty of Science, Xi'an Aeronautical University , Xi'an 710077 , China 1 College of Physical Science and Technology, Yangzhou University , Yangzhou 225009 , China We study the P-V criticality and phase transition in the extended phase space of anti-de Sitter (AdS) black holes in higher-dimensional de Rham, Gabadadze and Tolley (dRGT) massive gravity, treating the cosmological constant as pressure and the corresponding conjugate quantity is interpreted as thermodynamic volume. Besides the usual small/large black hole phase transitions, the interesting thermodynamic phenomena of reentrant phase transitions (RPTs) are observed for black holes in all d ≥ 6-dimensional spacetime when the coupling coefficients ci m2 of massive potential satisfy some certain conditions. - 1 Introduction Einstein’s general relativity (GR) is a relativistic theory of gravity where the graviton is a massless spin-2 particle [ 1– 3 ]. It is also the current description of gravitation in modern physics and has significant astrophysical implications. Nevertheless, whether there exist a consistent extension of GR by a mass term is a basic challenge of classical field theory, since the open questions such as the old cosmological constant problem and the origin of the late-time acceleration of the Universe remain behind the puzzles at the interface between gravity/cosmology and particle physics. In general, by adding generic mass terms for the gravitons on the given background usually brings about various instabilities for the gravitational theories, sometimes on the nonlinear level. A new nonlinear massive gravity theory was proposed by de Rham, Gabadadze and Tolley (dRGT) [ 4–6 ], where the Boulware–Deser ghost [ 7 ] was eliminated by introducing higher order interaction terms in the action. Until now, a nontrivial black hole solution with a Ricci flat horizon has been constructed by Vegh [ 8,9 ] in four-dimensional dRGT massive gravity. Later, spherically symmetric solutions were also addressed in [ 10–12 ], and the corresponding charged black hole solution was found in [13], including its bi-gravity extension [ 14,15 ]. Moreover, the charged AdS black hole solution in higher-dimensional dRGT massive gravity, and its corresponding thermodynamics and phase structure in the grand canonical and canonical ensembles were also presented in [16]. Ge et al. [ 17 ] examined the relations between dynamical instabilities and thermodynamic instabilities in the dRGT massive gravity. Recently, the study of thermodynamics in AdS black holes has been generalized to the extended phase space, where the cosmological constant is regarded as a variable and also identified with thermodynamic pressure [ 18,19 ] (1) P = − 8π = (d − 1)(d − 2) 16πl2 in geometric units G N = h¯ = c = k = 1. Here d stands for the number of spacetime dimensions and l denotes the AdS radius. In this case, the variation of the cosmological constant is included in the first law of black hole thermodynamics, which ensures the consistency between the first law of black hole thermodynamics and the Smarr formula. In [ 20 ], it was pointed out that the extended phase space can be interpreted as an RG-flow in the space of field theories, where isotherm curves codify how the number of degrees of freedom N (or the central charge c) runs with the energy scale. Moreover, the variation of cosmological constant could be corresponded to variation of number of the colors in Yang–Mills theory residing on the boundary spacetime [ 21,22 ]. In the extended phase space, the charged AdS black hole admits a more direct and precise coincidence between the first order small/large black holes (SBH/LBH) phase transition and Van der Waals liquidgas phase transition, and both systems share the same critical exponents near the critical point [23]. As a result, the analogy between the charged AdS black hole and the Van der Waals system becomes more complete. More discussions in various gravity theories can be found in [ 24–47 ]. In this direction, some investigations for thermodynamics of AdS black holes in the dRGT massive gravity have been generalized to the extended phase space [48–53], which revealed the existence of Van der Waals-like SBH/LBH phase transition. In addition, a link between the Van der Waals-like SBH/LBH phase transition and quasinormal modes (QNMs) has established in four [54] and higher-dimensional Reissner–Nordström AdS black hole [55], including time-domain profiles [56], and higher-dimensional charged black hole in the presence of Weyl coupling [57]. In terms of AdS/CFT, holographic entanglement entropy (HEE), Wilson loop, and two point correlation function also provide useful tools to probe the Van der Waals-like SBH/LBH phase transition [58–63]. Recently, Ref. [64] firstly recovered the existence of intermed (...truncated)