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 ${\lambda }_c\approx 0.6339$, while spontaneous scalarization solutions emerge for $\lambda >{\lambda }_c$. 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.