The Ginzburg-Landau Free Energy In Holographic Superconductor

The approach to study holographic superconductor with the Gauge/Gravity correspon-dence is a novel method to explore the strong coupled system, for instance, the research to high temperature superconductor. It triggers an instructive thought to tackle strong coupled system. Holographic superconductor system is composed of gravitational field; gauge field and complex scalar field. A lot of researches are done since the mechanism of holographic superconductor was posed by Gubser et.al. in2008, and many of numerical computations and simulations reveal the Ginzburg-Landau paradigm is obeyed near the critical temper-ature. However the analytic formulas of the order-parameter and Ginzburg-Landau free energy were not obtained.Our motivation is to analytically derive the Ginzburg-Landau free energy from holo-graphic superconductor action and seek the general expression of order-parameter. A co-nundrum of the coupled fields is there exist no analytic solution for their motion of equa-tions. However all information about condensate and phase transition are concealed within those equations. To tackle this difficult problem, we change the differential equation of gauge field into an integral form and obtain its iterative solution, substituting the iterative solution of gauge field in a specified ensemble, we switch the holographic superconduc-tor action into functional of scalar field. Performing a variation, we have a non-linear integro-differential equation of scalar field to describe its behavior deviating from the crit-ical temperature, where the asymptotic coefficients correspond to the one-point function on AdS boundary theory. On the other hand, at the critical point, the gauge field has an analytic solution and the EoM of the scalar field becomes a standard Sturm-Liouville prob- lem and this differential equaiont is a Fuchian equaiont with Five singularities, its leading eigenvalue is related to the critical temperature. We employ the complete functions from the Sturm-Liouville equation to expand the solution of the integro-differential equation and obtain the relation between the order-parameter and expansion coefficients. At last, the ana-lytic formulas of order-parameter and Ginzburg-Landau coefficients are obtained in (grand) canonical ensemble with conformal dimension one and two, respectively.We verified the consistency between our formulation and the fundamental thermody-namic relation. Utilizing our formulas, we calculated the scaling coefficients of order-parameter and Ginzburg-Landau coefficients in s-wave holographic superconductor. For the canonical ensemble part, the scaling coefficient has been computed via numerical simu-lation since2008, our results is in close good agreement with the results from computation. The difference of the scaling coefficients between the two thermodynamic ensembles is re-vealed manifestly from our formulas. By comparing with the situation in classically weak coupled BCS theory, we interpret this difference results from the characteristic of strong coupled system. Further, the Ginzburg-Landau free energy in inhomogenous space is stud-ied.