On the triplet of holographic phase transition

Abstract: We start from an Einstein $-$ Maxwell system coupled with a charged scalar field in Anti$-$de Sitter space$-$time. In the setup where the pressure $P$ is identified with the cosmological constant, the AdS black hole (BH) undergoes the phase transition from small to large BHs, which is similar to the transition from liquid to gas in the van der Waals theory. Based on this framework, we study the triplet of holographic superconducting states, consisting of ground state and two lowest excited states. Our numerical calculations show that the pressure variation in the bulk creates a mechanism in the boundary which causes changes in the physical properties of excited states, namely: a) when the pressure $ P $ is higher than the critical pressure ${P_c}$ ($ P > {P_c} $) of the phase transition from small to large BHs the ground state and the first excited state are superconducting states while the second excited state is the normal one. However, at lower pressure, $P \le P_c$, the ground state is solely the superconducting state. We conjecture that the precedent phenomena could take place when the scalar field in the bulk is replaced by other matter fields.