We present a class of charged black hole solutions in an (n + 2)-dimensional massive gravity with a negative cosmological constant, and study thermodynamics and phase structure of the black hole solutions both in grand canonical ensemble and canonical ensemble. The black hole horizon can have a positive, zero or negative constant curvature characterized by constant k. By using Hamiltonian approach, we obtain conserved charges of the solutions and find black hole entropy still obeys the area formula and the gravitational field equation at the black hole horizon can be cast into the first law form of black hole thermodynamics. In grand canonical ensemble, we find that thermodynamics and phase structure depends on the combination k − µ 2 /4 + c 2 m 2 in the four dimensional case, where µ is the chemical potential and c 2 m 2 is the coefficient of the second term in the potential associated with graviton mass. When it is positive, the Hawking-Page phase transition can happen, while as it is negative, the black hole is always thermodynamically stable with a positive capacity. In canonical ensemble, the combination turns out to be k + c 2 m 2 in the four dimensional case. When it is positive, a first order phase transition can happen between small and large black holes if the charge is less than its critical one. * E-mail: cairg@itp.ac.cn † E-mail: huyp@nuaa.edu.cn ‡ E-mail: panqiyuan@126.com § E-mail: ylzhang@itp.ac.cn 1 arXiv:1409.2369v1 [hep-th] Sep 2014In higher dimensional (n + 2 ≥ 5) case, even when the charge is absent, the small/large black hole phase transition can also appear, the coefficients for the third (c 3 m 2 ) and/or the fourth (c 4 m 2 ) terms in the potential associated with graviton mass in the massive gravity can play the same role as the charge does in the four dimensional case.

We investigate the behaviors of entanglement entropy in the holographical insulator/superconductor phase transition. We calculate the holographic entanglement entropy for two kinds of geometry configurations in a completely back-reacted gravitational background describing the insulator/superconductor phase transition. The non-monotonic behavior of the entanglement entropy is found in this system. In the belt geometry case, there exist four phases characterized by the chemical potential and belt width.

Using the non-relativistic hydrodynamic limit, we solve equations of motion for Einstein gravity and Gauss-Bonnet gravity with a negative cosmological constant within the region between a finite cutoff surface and a black brane horizon, up to second order of the non-relativistic hydrodynamic expansion parameter. Through the Brown-York tensor, we calculate the stress energy tensor of dual fluids living on the cutoff surface. With the black brane solutions, we show that for both Einstein gravity and Gauss-Bonnet gravity, the ratio of shear viscosity to entropy density of dual fluid does not run with the cutoff surface. The incompressible Navier-Stokes equations are also obtained in both cases. *