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

We address an issue of how to accurately include the self-energy effect of the screened electron-electron Coulomb interaction in 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 nonanalytic 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 different manners. 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.

Figure 1

Corrections to the Fermi velocity. The total correction with the RPA and $G_0{W}_0$ formulas (solid and open squares). The corrections $\mathrm{\Delta }{v}_{\chi }={\mathrm{\Delta }}_{\chi }$ and $\mathrm{\Delta }{v}_Z=k_{\mathrm{F}}{\mathrm{\Delta }}_Z$, respectively (open and solid circles). Data from Ref. [26] (crosses). The Quinn-Ferrell formula (28) (line).

Figure 2

Corrections to the inverse effective mass. See Fig. 1 for a description of the symbols.

Figure 3

Calculated normal-state self-energy for $r_{\mathrm{s}}=3.0$. (a) Overview of $\chi (k,i\omega )$ and (b) that along specific lines. (c) Overview of $\omega Z(k,i\omega )$ and (d) that along a specific line. The subtle oscillation in (d) is due to a numerical instability of the formula at $k\simeq k_{\mathrm{F}}, \left|\omega \right|>{\omega }_{\mathrm{thr}}$ (see Appendix pp2 for the definition of ${\omega }_{\mathrm{thr}}$).

Figure 4

Ratio of the largest eigenvalues of the linear kernel in Eq. (40) with and without the self-energy corrections. Approximate formulas are plotted with lines and points.