Continuous Hawking-Page transitions in Einstein-scalar gravity

We investigate continuous Hawking-Page transitions in Einstein's gravity coupled to a scalar field with an arbitrary potential in the weak gravity limit. We show that this is only possible in a singular limit where the black-hole horizon marginally traps a curvature singularity. Depending on the subleading terms in the potential, a rich variety of continuous phase transitions arise. Our examples include second and higher order, including the Berezinskii-Kosterlitz-Thouless type. In the case when the scalar is dilaton, the condition for continuous phase transitions lead to (asymptotically) linear-dilaton background. We obtain the scaling laws of thermodynamic functions, as well as the viscosity coefficients near the transition. In the limit of weak gravitational interactions, the bulk viscosity asymptotes to a universal constant, independent of the details of the scalar potential. As a byproduct of our analysis we obtain a one-parameter family of kink solutions in arbitrary dimension d that interpolate between AdS near the boundary and linear-dilaton background in the deep interior. The continuous Hawking-Page transitions found here serve as holographic models for normal-to superfluid transitions.