Critical behaviors of 3D black holes with a scalar hair

The principal focus of the present work concerns the critical behaviors of a class of three-dimensional (3D) black holes with a scalar field hair. Since the cosmological constant is viewed as a thermodynamic pressure and its conjugate quantity as a volume, we examine such properties in terms of two parameters B and a. The latters are related to the scalar field and the angular momentum, respectively. In particular, we give the equation of state predicting a critical universal number depending on the (B, a) moduli space. In the vanishing limit of the B parameter, we recover the usual perfect gas behavior appearing in the case of the non-rotating BTZ black hole. We point out that in a generic region of the (B, a) moduli space, the model behaves like a Van der Waals system.