Asymptotic upper bound to the Homes scaling relation in electron-phonon superconductors

The Homes scaling relation in superconductors relates the T=0 superfluid density to the normal-state DC electrical conductivity just above Tc. This linear relationship was originally proposed as a unique signature for high-Tc phenomena, yet it has been known since the 1960s that such linear scaling is present in dirty BCS superconductors. We find that Homes scaling may be present in electron-phonon superconductors in both the dirty and clean limits, and we derive a numerical value for the linear "Homes slope" at strong coupling. Our work sets a universal upper-bound for the Homes slope in all superconductors described by Eliashberg theory, regardless of the underlying bosonic propagator or the precise upper limit for the electron-phonon coupling strength.