Reentrant phase transitions in Einstein-Maxwell-scalar black holes

We consider how scalar fields affect the thermodynamic behavior of charged anti-de Sitter (AdS) black holes. We specifically investigate a class of ($3+1$)-dimensional exact hairy charged AdS black hole solutions to Einstein-Maxwell-scalar gravity, whose stable ground state and finite horizon area in the zero temperature limit make it of particular interest. We find that the reverse isoperimetric inequality is satisfied for this class and that there exists an intermediate range of the charge that admits reentrant phase behavior, the first example of this type of phase behavior in ($3+1$) dimensions in a consistent theory.