Various types of phase transitions in the AdS soliton background

We study the basic holographic insulator and superconductor phase transition in the AdS soliton background by generalizing the spontaneous breaking of a global U(1) symmetry to occur via St$\ddot{u}$ckelberg mechanism. We construct the soliton solutions with backreaction and examine the effects of the backreaction on the condensation of the scalar hair in the generalized St$\ddot{u}$ckelberg Lagrangian. We disclose rich physics in various phase transitions. In addition to the AdS soliton configuration, we also examine the property of the phase transition in the AdS black hole background.