The underlying black hole phase transitions in an Einstein-Maxwell-dilaton model with a holographic critical point

The Einstein-Maxwell-dilaton model exhibits a first-order phase transition curve that terminates at a holographic critical endpoint, offering intriguing insights into the phase diagram of the dual system living on the boundary. However, the specific instability within the underlying spacetime that triggers the formation of the hairy black hole remains somewhat obscure. This raises the question of whether one of the hairy black hole phases represents a superconducting or scalarized state and how the two distinct phases merge into a indistinguishable one at the critical point. This work investigates the associated black hole phase transition and the underlying instabilities by exploring a specific Einstein-Maxwell-dilaton model. The approach aims to provide transparent insights into the black hole phase transitions in the bulk. By introducing a nonminimal coupling between a massive real scalar field and a Maxwell field in a five-dimensional anti-de Sitter spacetime, we identify two types of scalarization corresponding to tachyonic instabilities in the ultraviolet and infrared regions. These distinct instabilities lead to a first-order phase transition between two phases in the $\mu-T$ phase diagram. Furthermore, this first-order transition terminates at a critical point, beyond which the curve turns back, and the transition becomes a numerically elusive third-order one. Although one does not encounter a critical endpoint, the model still offers a consistent interpretation for the observed {\it cross-over} in the low baryon density region. We analyze the thermodynamic properties of the scalarized hairy black holes and discuss the implications of our findings.