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 c i m 2 of massive potential satisfy some certain conditions.

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We found the eigenvalues of the transfer matrices for the 1-D Hubbard model and for the coupled XY model with twisted boundary condition by using the analytic Bethe Ansatz method. Under a particular condition the two models have the same Bethe Ansatz equations. We have also proved that the periodic 1-D Hubbard model is exactly equal to the coupled XY model with nontrivial twisted boundary condition at the level of hamiltonians and transfer matrices.

In this work, we utilize the quasinormal modes (QNMs) of a massless scalar perturbation to probe the Van der Waals-like small and large black holes (SBH/LBH) phase transition of charged topological Anti-de Sitter (AdS) black holes in four-dimensional massive gravity.We find that the signature of this SBH/LBH phase transition is detected in the isobaric as well as in the isothermal process. This further supports the idea that the QNMs can be an efficient tool to investigate the thermodynamical phase transition. PACS numbers: 04.50.Kd, 04.70.-s, 04.25.D-I. INTRODUCTION Einstein's general relativity introduces gravitons as massless spin-2 particles [1-3]. However, understanding the quantum behavior of gravity could be related to the possible mass of the graviton.This Einstein theory, modified at large distances in massive gravity provides a possible explanation for the accelerated expansion of the Universe that does not require any dark energy. Actually, the massive gravity and its extensions, such as bimetric gravity, can yield cosmological solutions which do display late-time acceleration in agreement with observations [4][5][6][7]. Very recently, the LIGO collaboration reporting the discovery of gravitational wave asserted that [8] "assuming a modified dispersion relation for gravitational waves, our observations constrain the Compton wavelength of the graviton to be λ g > 10 13 km, which could be interpreted as a bound on the graviton mass m g < 1.2 × 10 −22 eV /c 2 ". In order to have massive graviton, the first attempt for constructing a massive theory was in the work of Fierz and Pauli [9] which was done in the context of linear theory.Unfortunately this theory possesses so-called van Dam, Veltman and Zakharov discontinuity prob- *

The publication of this article was funded by SCOAP 3 .Motivated by recent developments of black hole thermodynamics in de Rham, Gabadadze, and Tolley (dRGT) massive gravity, we study the critical behaviors of topological Anti-de Sitter (AdS) black holes in the presence of Born-Infeld nonlinear electrodynamics.Here the cosmological constant appears as a dynamical pressure of the system and its corresponding conjugate quantity is interpreted as thermodynamic volume. This shows that, besides the Van der Waals-like SBH/LBH phase transitions, the socalled reentrant phase transition (RPT) appears in four-dimensional space-time when the coupling coefficients 2 of massive potential and Born-Infeld parameter satisfy some certain conditions. In addition, we also find the triple critical points and the small/intermediate/large black hole phase transitions for = 5.

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It is found by Cai and Su that the interaction between dark energy and cold dark matter is likely to change the sign during the cosmological evolution. Motivated by this, we suggest a new form of interaction between dark energy and dark matter, which changes from negative to positive as the expansion of our universe changes from decelerated to accelerated. We find that the interacting model is consistent with the second law of thermodynamics and the observational constraints. And, we also discuss the unified adiabatic-squared sound speed of the model. : 95.36.+x, 98.80.-k, 95.35+d PACS