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Scaling of Coulomb pseudo-potential in s-wave narrow-band superconductors

The Coulomb pseudo-potential $μ^*$ is extracted by fitting the numerically calculated transition temperature $T_c$ of the Eliashberg-Nambu equation which is extended to incorporate the narrow-band effects, that is, the vertex correction and the frequency dependence of the screened Coulomb interaction. It is shown that even for narrow-band superconductors, where the fermi energy $ ε_F$ is comparable with the phonon frequency $ ω_{ph}$, the Coulomb pseudo-potential is a pertinent parameter, and is still given by $μ^* = μ/[1+μ\ln(ε_F/ω_{ph})] $, provided $ω_{ph}$ is appropriately scaled.