Phase Transition in Holographic Superconductor

A gravitational description of a holographic superconductor in the sense of AdS/CFT correspondence is studied. Superconductivity is due to condensation of charged scalar operator caused by a broken global U(1) symmetry on the boundary of an Anti de Sitter (AdS) black hole. This mechanism translates to an instability of the AdS. A scalar field in the bulk acquires a non-trivial vacuum expectation value. The bulk equations of motions for the gravity theory are solved numerically to find solutions of the charged scalar. Superconducting phase transitions emerges around a critical temperature when either a chemical potential or charge density of the AdS black hole is fixed. These correspond to canonical and grand canonical ensembles, respectively. The results for both cases are similar, in the sense that we find second order phase transitions in either case. The superconducting phase transition is second order at the critical temperature. The temperatures when density is one and chemical potential is negative, are numerically computed to be 0.2683 and 0.3013, respectively. Also, the critical exponent is found to be 0.449 for the fixed density and 0.476 for the fixed chemical potential.