Holographic phase transition for a black hole in the background with dark energy

Abstract

In the background with dark energy, we investigate the phase structure of thermodynamic entropy and entanglement entropy. We investigate respectively the effect of the charge and state parameters of the dark energy on the phase structure of the black hole. For a fixed state parameter of dark energy, as the value of the charge increases, the phase structures of the thermodynamical entropy and the entanglement entropy are similar as that of the Van der Waals phase transition. That is, the black hole undergoes the first order phase transition and second order phase transition successively and reaches to a stable state at last. For a fixed charge, as the state parameter of dark energy increases, the phase structures of the thermodynamical entropy is not similar to the entanglement entropy. Especially, the tendency of the change of entanglement entropy with respect to the state parameter is opposite to the thermodynamic entropy. The same is that the black hole undergoes the first order phase transition and second order phase transition successively as the state parameter of dark energy increases. For the thermodynamics entropy and entanglement entropy, we find Maxwells equal area law holds for the unstable region of the first order phase transition, and the critical exponent of the heat capacity is 2/3 near the critical points of the second order phase transition.