Observing various phase transitions in the holographic model of superfluidity

We study the gravity duals of supercurrent solutions in the AdS black hole background with general phase structure to describe both the first and the second order phase transitions at finite temperature in strongly interacting systems. We argue that the conductivity and the pair susceptibility can be possible phenomenological indications to distinguish the order of phase transitions. We extend our discussion to the AdS soliton configuration. Different from the black hole spacetime, in the probe limit the first order phase transition cannot be brought by introducing the spatial component of the vector potential of the gauge field in the AdS soliton background.