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

Various types of phase transitions in the AdS soliton background