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

We reassess the validity of the Migdal-Eliashberg (ME) theory for coupled electron-phonon systems for large couplings lambda. Although model calculations have found that the ME theory breaks down for lambda similar to 0.5, it is routinely applied for 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 lambda. This provides quantitative theoretical support for the applicability of the ME theory to strong-coupling conventional superconductors.