Revisiting the homogeneous electron gas in pursuit of the properly normed ab initio Eliashberg theory

Abstract: We address an issue of how to accurately include the self energy effect of the screened electron-electron Coulomb interaction in the phonon-mediated superconductors from first principles. In the Eliashberg theory for superconductors, self energy is usually decomposed using the $2\times 2$ Pauli matrices in the electron-hole space. We examine how the diagonal ($\sigma_{0}$ and $\sigma_{3}$) components resulting in the quasiparticle correction to the normal state, $Z$ and $\chi$ terms, behave in the homogeneous electron gas in order to establish a norm of treating those components in real metallic systems. Within the $G_{0}W_{0}$ approximation, we point out that these components are non-analytic near the Fermi surface but their directional derivatives and resulting corrections to the quasiparticle velocity are nevertheless well defined. Combined calculations using the $G_{0}W_{0}$ approximation and Eliashberg equations show us that the effective mass and pairing strength strikingly depend on both $Z$ and $\chi$, in a different manner. The calculations without the numerically demanding $\chi$ term is thus shown to be incapable of describing the homogeneous electron gas limit. This result poses a challenge to accurate first-principles Eliashberg theory.