Quantiative reliability of the Migdal-Eliashberg theory for strong coupling superconductors

The Migdal-Eliashberg (ME) theory for strong electron-phonon coupling and retardation effects of the Morel-Anderson type form the basis for the quantitative understanding of conventional superconductors. The validity of the ME theory for values of the electron-phonon coupling strength $\lambda>1$ has been questioned by model studies. By distinguishing bare and effective parameters, and by comparing the ME theory with the dynamical mean field theory (DMFT), we clarify the range of applicability of the ME theory. Specifically, we show that ME theory is very accurate as long as the product of effective parameters, $\lambda \omega_{\rm ph}/D$, where $\omega_{\rm ph}$ is an appropriate phonon scale and $D$ an electronic scale, is small enough [1]. The effectiveness of retardation effects is usually considered based on the lowest order diagram in the perturbation theory. We analyze these effects to higher order and find modifications to the usual result for the Coulomb pseudo-potential $\mu^*$. Retardation effects are weakened due to a reduced effective bandwidth. Comparison with the non-perturbative DMFT corroborates our findings~[2]. \\[4pt] [1] J Bauer, J E Han, and O Gunnarsson, Phys. Rev. B. 84, 184531 (2011).\\[0pt] [2] J Bauer, J E Han, and O Gunnarsson, in preparation (2011).