Analysis of exact vertex function for improving on the GWΓscheme for first-principles calculation of electron self-energy

Abstract

In order to propose a sophisticated scheme for the self-consistent calculation of the electron self-energy $\ensuremath{\Sigma}$, a detailed analysis of the analytical structure of the three-point vertex function $\ensuremath{\Gamma}$ is made with full respect for the Ward identity from the perspective of Fermi-liquid theory. Our scheme may be regarded as an improvement on the gauge-invariant self-consistent approximation to the exact theory for obtaining $\ensuremath{\Sigma}$ as a fixed point of the self-energy revision operator $\mathcal{F}$, indicating an intrinsically nonperturbative approach applicable to both Fermi and Tomonaga-Luttinger liquids in a unified manner, but it may also be considered as providing a general framework for constructing an accurate functional form for $\ensuremath{\Gamma}$ in the GW$\ensuremath{\Gamma}$ method for the first-principles calculation of $\ensuremath{\Sigma}$. Our result on the momentum distribution function in the homogeneous electron gas is compared with the one recently obtained by the reptation Monte Carlo method.