Scalar-hairy AdS Black Hole in the Einstein-Maxwell-Scalar Theory: first-order phase transition with a critical point

In asymptotically anti-de Sitter (AdS) spacetime, we consider a real massiver scalar field in the Einstein-Maxwell-scalar (EMS) model and examine both scalar-hairy black hole solutions induced by the nonminimal coupling to the Maxwell field and tachyonic-hairy solutions driven by the scalar potential. When the scalar potential vanishes, scalar-hairy black holes emerge with profiles and properties similar to those observed in flat spacetime. The presence of the scalar potential additionally induces tachyonic-hairy solutions, leading to the coexistence of these two distinct hairy phases in different regions of the parameter space. The phase diagram reveals a first-order phase transition line between the tachyonic-hairy and scalar-hairy phases, originating at a critical point in the extreme temperature and chemical potential regime. Our detailed analysis shows that this phase transition is directly associated with the self-overlap region of the scalar-hairy phase and its start point. Moreover, increasing the coupling strength $\lambda$ shifts the critical point to higher temperature and chemical potential.