Novel phase transition in charged dilaton black holes

We disclose a novel phase transition in black hole physics by investigating thermodynamics of charged dilaton black holes in an extended phase space where the charge of the black hole is regarded as a fixed quantity. Along with the usual critical (second-order) as well as the first-order phase transitions in charged black holes, we find that a finite jump in Gibbs free energy is generated by dilaton-electromagnetic coupling constant, $\alpha$, for a certain range of pressure. This novel behavior indicates a small/large black hole \emph{zeroth-order} phase transition in which the response functions of black holes thermodynamics diverge e.g. isothermal compressibility. Such zeroth-order transition separates the usual critical point and the standard first-order transition curve. We show that increasing the dilaton parameter($\alpha$) increases the zeroth-order portion of the transition curve. Additionally, we find that the second-order (critical) phase transition exponents are unaffected by the dilaton parameter, however, the condition of positive critical temperature puts an upper bound on the dilaton parameter ($\alpha<1$).