Homes' law in holographic superconductor with linear-$T$ resistivity

Homes' law, $\rho_{s} = C \, \sigma_{DC} \, T_{c}$, is a universal relation of superconductors between the superfluid density $\rho_{s}$ at zero temperature, the critical temperature $T_{c}$ and the electric DC conductivity $\sigma_{DC}$ at $T_c$. Experimentally, Homes' law is observed in high $T_c$ superconductors with linear-$T$ resistivity in the normal phase, giving a material independent universal constant $C$. By using holographic models related to the Gubser-Rocha model, we investigate how Homes' law can be realized together with linear-$T$ resistivity in the presence of momentum relaxation. We find that strong momentum relaxation plays an important role to exhibit Homes' law with linear-$T$ resistivity.
Comment: v1: 24 pages, 9 figures; v2: presentation improved