Quantitative reliability study of the Migdal-Eliashberg theory for strong electron-phonon coupling in superconductors

Abstract

We reassess the validity of the Migdal-Eliashberg (ME) theory for coupled electron-phonon systems for large couplings $\ensuremath{\lambda}$. Although model calculations have found that the ME theory breaks down for $\ensuremath{\lambda}\ensuremath{\sim}0.5$, it is routinely applied for $\ensuremath{\lambda}>1$ to strong-coupling superconductors. To resolve this discrepancy, it is important to distinguish between bare parameters, used as input in models, and effective parameters, derived from experiments. We show explicitly that ME gives accurate results for the critical temperature and the spectral gap for large effective $\ensuremath{\lambda}$. This provides quantitative theoretical support for the applicability of the ME theory to strong-coupling conventional superconductors.