Theoretical study of phonon-mediated superconductivity beyond Migdal-Eliashberg approximation and Coulomb pseudopotential

TL;DR

This paper develops a self-consistent Dyson-Schwinger equation approach to study phonon-mediated superconductivity, going beyond traditional approximations to better understand the interplay of electron-phonon and Coulomb interactions, with applications to FeSe/SrTiO3.

Contribution

It introduces a novel self-closed Dyson-Schwinger equation framework that includes all vertex corrections and interaction effects, improving the theoretical modeling of superconductivity.

Findings

Vertex corrections significantly affect $T_c$.

Interplay between phonon attraction and Coulomb repulsion influences superconductivity.

Application to FeSe/SrTiO$_{3}$ shows strong impact on $T_c$.

Abstract

In previous theoretical studies of phonon-mediated superconductors, the electron-phonon coupling is treated by solving the Migdal-Eliashberg equations under the bare vertex approximation, whereas the effect of Coulomb repulsion is incorporated by introducing one single pseudopotential parameter. These two approximations become unreliable in low carrier-density superconductors in which the vertex corrections are not small and the Coulomb interaction is poorly screened. Here, we shall go beyond these two approximations and employ the Dyson-Schwinger equation approach to handle the interplay of electron-phonon interaction and Coulomb interaction in a self-consistent way. We first derive the exact Dyson-Schwinger integral equation of the full electron propagator. Such an equation contains several unknown single-particle propagators and fermion-boson vertex functions, and thus seems to be intractable. To solve this difficulty, we further derive a number of identities satisfied by all the relevant propagators and vertex functions and then use these identities to show that the exact Dyson-Schwinger equation of electron propagator is actually self-closed. This self-closed equation takes into account not only all the vertex corrections, but also the mutual influence between electron-phonon interaction and Coulomb interaction. Solving it by using proper numerical methods leads to the superconducting temperature $T_{c}$ and other quantities. As an application of the approach, we compute the $T_{c}$ of the interfacial superconductivity realized in the one-unit-cell FeSe/SrTiO$_{3}$ system. We find that $T_{c}$ can be strongly influenced by the vertex corrections and the competition between phonon-mediated attraction and Coulomb repulsion.