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

The Coulomb pseudo-potential ${\mu }^{\ast }$ 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 ${\omega }_{ph}$, the Coulomb pseudo-potential is a pertinent parameter, and is still given by $\mu^* = \mu/[1+\mu \ln(\epsilon_F/\omega_{ph})] $,provided$\omega_{ph}$ is appropriately scaled.