Towards exact solutions for the superconducting $T_c$ induced by electron-phonon interaction

TL;DR

This paper develops a nonperturbative Dyson-Schwinger equation approach that includes all vertex corrections to accurately determine the superconducting transition temperature ($T_c$) beyond traditional approximations, applicable to both weak and strong coupling regimes.

Contribution

It introduces a rigorous, self-closed integral equation for the electron propagator that accounts for all vertex corrections, advancing the theoretical understanding of superconductivity beyond Migdal-Eliashberg theory.

Findings

Successfully applied to high-$T_c$ FeSe/SrTiO$_3$ superconductors.

Provides a numerical method for calculating $T_c$ with full vertex corrections.

Works effectively across different coupling strengths.

Abstract

Electron-phonon interaction plays an important role in metals and can lead to superconductivity and other instabilities. Previous theoretical studies on superconductivity are largely based on the Migdal-Eliashberg theory, which neglects all the vertex corrections to electron-phonon coupling and breaks down in many unconventional superconductors. Here, we go beyond the Migdal-Eliashberg approximation and develop a nonperturbative Dyson-Schwinger equation approach to deal with the superconducting transition. Remarkably, we take into account all the vertex corrections by solving two coupled Ward-Takahashi identities derived from two global U(1) symmetries and rigorously prove that the fully renormalized electron propagator satisfies a self-closed integral equation that is directly amenable to numerical computations. Our approach works equally well in the weak and strong coupling regimes and provides an efficient method to determine superconducting $T_c$ and other quantities. As an application, our approach is used to investigate the high-$T_c$ superconductivity in one-unit-cell FeSe/SrTiO$_3$.