Eliashberg Theory in the Weak Coupling Limit

TL;DR

This paper provides highly accurate numerical solutions to Eliashberg theory in the weak coupling limit, clarifying its relationship with BCS theory and extending analytical results to first order in the coupling constant.

Contribution

It offers precise numerical solutions and extends analytical results, strengthening the understanding of Eliashberg theory's weak coupling limit.

Findings

Numerical solutions confirm the weak coupling limit validity.

Extended analytical results to first order in coupling constant.

Clarified the distinction between Eliashberg and BCS theories.

Abstract

Eliashberg theory provides a theoretical framework for understanding the phenomenon of superconductivity when pairing between two electrons is mediated by phonons, and retardation effects are fully accounted for. BCS theory is often viewed as the weak coupling limit of Eliashberg theory, in spite of a handful of papers that have pointed out that this is not so. Here we present very accurate numerical solutions in the weak coupling limit to complement the existing analytical results, and demonstrate more convincingly the validity of this limit by extending the analytical results to first order in the coupling constant.