Homes' law in Holographic Superconductor with Q-lattices

Understand

Homes' law, $\rho_s = C \sigma_{\mathrm{DC}} T_c$, is an empirical law satisfied by various superconductors with a material independent universal constant $C$, where $\rho_{s}$ is the superfluid density at zero temperature, $T_c$ is the critical temperature, and $\sigma_{\mathrm{DC}}$ is the electric DC conductivity in the normal state close to $T_c$.

• We study Homes' law in holographic superconductor with Q-lattices and find that Homes' law is realized for some parameter regime in insulating phase near the metal-insulator transition boundary, where momentum relaxation is strong.
• In computing the superfluid density, we employ two methods: one is related to the infinite DC conductivity and the other is related to the magnetic penetration depth.
• With finite momentum relaxation both yield the same results, while without momentum relaxation only the latter gives the superfluid density correctly because the former has a spurious contribution from the infinite DC conductivity due to translation invariance.