Holographic Superconducting Phase Transition

The AdS/CFT correspondence is a most remarkable result from string theory, which relates string theory on asymptotically anti de Sitter spacetimes to a conformal field the-ory on the boundary and provided a description of the superconductor through a gravity dual, called holographic superconductor.The scalar field condenses and the holographic superconductors system undergo a superconducting phase transition below a critical temperature.The d dimensional field theory is dual to a d+1dimensional gravitational description of the system, the transition of which to superconductivity is observed as a classical instability of a black hole in anti-de Sitter (AdS) space against perturbations by a charged scalar field ψ. The scalar field condense when the black hole has a tem-perature below the Hawking temperature T=Tc. The gravity models have attracted considerable interest for their potential applications to the condensed matter physics.In this work, we give a brief introduction of the holographic superconductor defini-tion, importance and status. Also we give some result of the Stuckelberg superconduc-tor model got by others in AdS black hole background. We have put this Stuckelberg superconductor model established in black hole into soliton spacetime and give more talk for the model in black hole background. We have studed which scalor operator is incapable of describe the phase transition. The result in the first two sections are got by others and the third and fourth section are our own results. We show them in the following in detail:In the first section, we introduce the most simple holographic superconductor model and get the solution with numerical method. We talk about the known result, the method usually used and the importance of this model.In the second section, we include the matter field’s backreaction to the spacetime. We discuss the holographic superconductor model in black hole to study the instability of the system with different method. When extend the superconductor model to soliton spacetime, we give some known result. In general, the phase transition is second order. But it can be first order in the Stuckelberg model, which is established by considering the higher correction terms of the scaler field. We give some progress in Stuckelberg model. In the third section, considering the high correction of the scaler field we extend the Stuckelberg model of black hole to soliton backgroud. The known result with out including the high correction of the scaler field tells us that strong backreaction in soliton spacetinme can trigger first order phase transition. When the backreaction is weak, the phase transition is second order. In the Stuckelberg model in soliton, we find the high correction of the scaler field can introduce first order phase transition. In the probe limit, this conclusion still holds. In the case of black hole, we study the case when the backreaction is very heavy. We find when the backreaction parameter is above some critical value, the backreaction’s effect on the phase transition is very different from the case of weak backreaction.In the fourth section, we study which operator is more capable of describe the phase transition. The former results are concluded by study the second order phase transition. Here, in our superfluid model, we distinguish there two operators by first order phase transition. we conclude the second operator is more physical in the4dimensional AdS black hole by consider the relation of condensation gap and the critical temperature. This is in accedence with former result got from second order phase transition.In the last section, we give a summery of the work we have done.