The Ginzburg-Landau Theory of a Holographic Superconductor

The general Ginzburg-Landau formulation of a holographic superconductor is developed near the transition temperature in the probe limit for two kinds of conformal dimension. Below the transition temperature, $T<T_c$, the order-parameter scales with $\sqrt{1-\frac{T}{T_c}}$ as expected. The G-L free energy in grand caonnical ensemble and canonical ensemble are derived and the gradient term is studied. Furthermore this scaling coefficient of order-parameter takes different values in the grand canonical ensemble and the canonical ensemble, suggesting the strong coupling nature of the boundary field theory of the superconductiviy.