Critical behavior of charged dilaton black holes in AdS space

We revisit critical behaviour and phase structure of charged anti-deSitter (AdS) dilaton black holes for arbitrary values of dilaton coupling $\alpha$, and realize several novel phase behaviour for this system. We adopt the viewpoint that cosmological constant (pressure) is fixed and treat the charge of the black hole as a thermodynamical variable. We study critical behaviour and phase structure by analyzing the phase diagrams in $T-S$ and $q-T$ planes. We numerically derive the critical point in terms of $\alpha$ and observe that for $\alpha =1$ and $\alpha \ge \sqrt{3}$, the system does not admit any critical point, while for $0<\alpha <1$, the critical quantities are not significantly affected by $\alpha$. We find that unstable behavior of the Gibbs free energy for $q<q_c$ exhibits a $\mathit{\text{first}}\mathit{\text{order}}$ (discontinuous) phase transition between small and large black holes for $0\le \alpha <1$. For $1<\alpha <\sqrt{3}$ and $q>q_c$, however, a novel first order phase transition occurs between small and large black hole, which has not been observed in the previous studies on phase transition of charged AdS black holes.