Uniform electron benchmark for the first-principles $GW_{0}$-Eliashberg theory

TL;DR

This paper develops a comprehensive first-principles benchmark for superconductivity calculations by analyzing the Eliashberg equations with full Coulomb and phonon interactions in the uniform electron gas, revealing effects on superconducting properties.

Contribution

It introduces a detailed numerical approach for solving Eliashberg equations with the $GW_{0}$ approximation, accounting for full momentum and frequency dependence, providing a benchmark for first-principles superconductivity calculations.

Findings

Electron-phonon interactions significantly influence the effective mass and spectral weight.

Plasmon effects can enhance or suppress phonon-mediated superconductivity depending on electron density.

Results align with density functional theory, validating the approach as a benchmark.

Abstract

We investigate the numerical behavior of the Eliashberg equations for phonon-mediated superconductivity, incorporating normal-state self-energy calculations within the consistent $GW_{0}$ approximation. We account for the full wavenumber and frequency dependences of both the screened Coulomb interaction and phonon-mediated attraction. We present results for the prototypical uniform electron gas system with model Einstein phonons at temperatures of a few kelvin. At extremely low temperatures, we efficiently execute the required convolutions of Green's functions and interactions in Matsubara frequency and wavenumber using intermediate representation and Fourier convolution techniques. In particular, we elucidate the interplay between electron-phonon $\omega$-mass and $k$-mass renormalizations of the electronic self-energy in determining the normal-state effective mass, spectral weight and the superconducting transition temperature. The electron density regimes where the plasmon effect enhances or suppresses the phonon-mediated superconductivity on top of the static Coulomb effect are revealed. We compare our comprehensive Eliashberg calculation results with those from density functional theory for superconductors, where the functionals have been constructed with reference to Eliashberg theory. Our model, methods, and results provide a valuable benchmark for first-principles superconducting calculations that treat screened Coulomb interaction effects non-empirically.