Phase Structure of Holographic Superconductors With Spontaneous Scalarization

Within the holographic framework, we investigate the Einstein-scalar-Gauss-Bonnet theory coupled with a Maxwell field in an asymptotically anti-de Sitter (AdS) spacetime. Our findings reveal that holographic superconductor solutions exist when the Gauss-Bonnet coupling constant is below the critical value (Formula presented.), while spontaneous scalarization solutions emerge for (Formula presented.). This raises the question whether the hairy black holes triggered by different mechanisms are smoothly joined by a phase transition or whether these are actually identical solutions. To examine this transition in greater detail, we constructed a phase diagram in terms of temperature and chemical potential. By explicitly evaluating the Gibbs free energy and its derivatives, we demonstrate that a smooth yet first-order phase transition occurs between the two hairy solutions. In particular, a thermodynamic process can be devised in which a superconducting black hole transitions into a scalarized black hole by raising or lowering the temperature.