Full-bandwidth Eliashberg theory of superconductivity beyond Migdal's approximation

TL;DR

This paper numerically solves the full-bandwidth, anisotropic Eliashberg equations including vertex corrections to better understand non-adiabatic effects on superconductivity, revealing their variable impact depending on system parameters.

Contribution

It introduces a fully numerical solution of non-adiabatic Eliashberg equations with vertex corrections for a one-band system, surpassing previous approximations.

Findings

Non-adiabatic effects on the superconducting gap vary with system parameters.

Vertex corrections can positively, negatively, or negligibly influence superconductivity.

The study highlights the importance of beyond-approximation methods for accurate superconductivity modeling.

Abstract

We solve the anisotropic, full-bandwidth and non-adiabatic Eliashberg equations for phonon-mediated superconductivity by fully including the first vertex correction in the electronic self-energy. The non-adiabatic equations are solved numerically here without further approximations, for a one-band model system. We compare the results to those that we obtain by adiabatic full-bandwidth, as well as Fermi-surface restricted Eliashberg-theory calculations. We find that non-adiabatic contributions to the superconducting gap can be positive, negative or negligible, depending on the dimensionality of the considered system, the degree of non-adiabaticity, and the coupling strength. We further examine non-adiabatic effects on the transition temperature and the electron-phonon coupling constant. Our treatment emphasizes the importance of overcoming previously employed approximations in estimating the impact of vertex corrections on superconductivity and opens a pathway to systematically study vertex correction effects in systems such as high-$T_c$, flat band and low-carrier density superconductors.