Dependence of the Critical Temperature and Disorder in Holographic Superconductors on Superfluid Density

Recent experiments strongly indicate deep connections between transports of strange metal and high-$T_c$ superconductors. For instance, it is known that the dependence of the zero-temperature phase stiffness on the critical superconducting temperature becomes linear in underdoped materials. In this paper, we investigate meticulously the relation between the phase stiffness and the critical superconducting temperature for the Gubser-Rocha holographic superconductor model in the probe limit. The superfluid density (or phase stiffness) can be extracted from the low-frequency dependence of the AC conductivity. More importantly, we find linear dependence of zero-temperature superfluid density on the critical superconducting temperature, which bears a similarity to a recent experiment in a film cuprate. In addition, we also provide some approximate formulas for the critical temperatures and the AC conductivity.