Quantitative reliability of Migdal-Eliashberg theory for strong electron-phonon coupling

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

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