One dimensional s-wave holographic superconductor with supercurrent

We study the one dimensional s-wave holographic superconductor by turning on the vector potential $A_x$ in the bulk, which behaves as $A_x=A_x^{(0)} \ln z+ A_x^{(1)}$ on the boundary. By solving the model with fixed $A_x^{(0)}$, we find that if we identify the $A_x^{(0)}$ with the supercurrent $j_x$ of the holographic superconductor, the results agree with the Gindzburg- Landau theory. For example, $A_x^{(0)}$ will break the superconductivity, and the critical value of $A_x^{(0)}$ is proportional to $(T_c-T)^{3/2}$.