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

TL;DR

This paper demonstrates that Migdal-Eliashberg theory remains quantitatively reliable for predicting critical temperature and spectral gap in strong coupling superconductors when considering effective parameters, despite previous assumptions of its breakdown.

Contribution

It clarifies the applicability of ME theory for large effective electron-phonon coupling, reconciling model discrepancies with experimental relevance.

Findings

ME theory accurately predicts critical temperature for large effective λ

Spectral gap calculations remain reliable at strong coupling

Distinction between bare and effective parameters is crucial

Abstract

We reassess the validity of Migdal-Eliashberg (ME) theory for coupled electron-phonon systems for large couplings $\lambda$. Although model calculations have found that ME theory breaks down for $\lambda\sim 0.5$, it is routinely applied for $\lambda>1$ 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 $\lambda$. This provides quantitative theoretical support for the applicability of ME theory to strong coupling conventional superconductors.