Reentrant phase transition of Born-Infeld-AdS black holes

Abstract: We investigate thermodynamic phase structure and critical behaviour of Born-Infeld (BI) black holes in an anti-de Sitter (AdS) space, where the charge of the system can vary and the cosmological constant (pressure) is fixed. We find that the BI parameter crucially affects the temperature of the black hole when the horizon radius, $r_{+}$, is small. We observe that depending on the value of the nonlinear parameter, $\beta $, BI-AdS black hole may be identified as RN black hole for $Q\geq Q_{m}$, and Schwarzschild-like black hole for $Q<Q_{m}$, where $Q_{m}=1/\left(8\pi \beta \right) $ is the \textit{marginal} charge. We analytically calculate the critical point ($% Q_c,T_c, r_{+c}$) by solving the cubic equation and study the critical behaviour of the system. We also explore the behavior of Gibbs free energy for BI-AdS black hole. We find out that the phase behaviour of BI-AdS black hole depends on the charge $Q$. For $Q>Q_{c}$, the Gibbs free energy is single valued and the system is locally stable ($C_{Q}>0$), while for $% Q<Q_{c}$, it becomes multivalued and $C_{Q}<0$. In the range of $% Q_{z}<Q<Q_{c}$, a first order phase transition occurs between small black hole (SBH) and large black hole (LBH). Interestingly enough, in the range of $Q_{t}\leq Q\leq Q_{z}$, a reentrant phase transition occurs between intermediate (large) black hole, SBH and LBH in Schwarzschild-type region. This means that in addition to the first order phase transition which separates SBH and LBH, a finite jump in Gibbs free energy leads to a \textit{% zeroth order} phase transition between SBH and intermediate black hole (LBH) where initiates from $Q=Q_{z}$ and terminates at $Q=Q_{t}$.