Universal scaling relations in electron-phonon superconductors

In many-body physics, scaling relations often serve as a robust signature of universal physics independent of the underlying microscopic details. Perhaps most notably, scaling relations between observables in the superconducting ground state and the finite-temperature normal state have been explored as a means of understanding the pairing mechanism behind several prominent high-Tc superconductors. In this work, we investigate the scaling relationship between the T=0 superfluid density and the normal-state conductivity at T=Tc for a class of weakly- and strongly-coupled electron-phonon superconductors. Using the formalism of Eliashberg theory, we analytically and numerically show that a superconductor may support the aforementioned scaling relation by virtue of strong electron-phonon coupling. This work sheds light on how such scaling relations connecting the two phases have wider applicability than previously considered.