Supercurrent: Vector hair for an AdS black hole

Hartnoll, Herzog, and Horowitz [Phys. Rev. Lett. 101, 031601 (2008).] discuss a holographic black hole solution which exhibits a superconductorlike transition. In the superconducting phase the black holes show infinite DC conductivity. This gives rise to the possibility of deforming the solutions by turning on a time independent current (supercurrent), without any electric field. This type of deformation does not exist for normal (nonsuperconducting) black holes, due to the no-hair theorems. In this paper we have studied such a supercurrent solution and the associated phase diagram. Interestingly, we have found a ``special point'' (critical point) in the phase diagram where the second order superconducting phase transition becomes first order. Supercurrent in superconducting materials is a well studied phenomenon in condensed matter systems. We have found some qualitative agreement with known results.