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Homes' law in Holographic Superconductor with Q-lattices

Homes' law, $ρ_s = C σ_{\mathrm{DC}} T_c$, is an empirical law satisfied by various superconductors with a material independent universal constant $C$, where $ρ_{s}$ is the superfluid density at zero temperature, $T_c$ is the critical temperature, and $σ_{\mathrm{DC}}$ 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.