Critical behaviors of black holes in the Einstein-Maxwell gravity with conformal anomaly

Abstract: We study $P-V$ critical behavior of 4-dimensional AdS black hole in the Einstein-Maxwell gravity with conformal anomaly by treating the cosmological constant as a variable related to the thermodynamic pressure. It shows that there is no phase transition if taking $k=0$ or -1. When the charge $q_1$ of conformal field and the coefficient $\alpha$ satisfy a certain relation, the Van der Waal like phase transition for the spherical black hole can occur in case of the temperature is lower than the small critical temperature or higher than the large one. We also evaluate the critical exponents of the phase transitions and find that the thermodynamic exponents associated with this 4-dimensional AdS black hole coincide with those of the van der Waals fluid.