GWapproximations and vertex corrections on the Keldysh time-loop contour: Application for model systems at equilibrium

Abstract

We study the effects of self-consistency and vertex corrections on different $GW$-based approximations for model systems of interacting electrons. For dealing with the most general case, we use the Keldysh time-loop contour formalism to evaluate the single-particle Green's functions. We provide the formal extension of Hedin's $GW$ equations for the Green's function in the Keldysh formalism. We show an application of our formalism to the plasmon model of a core electron within the plasmon-pole approximation. We study in detail the effects of the diagrammatic perturbation expansion of the core-electron/plasmon coupling on the spectral functions in the so-called S model. The S model provides an exact solution at equilibrium for comparison with the diagrammatic expansion of the interaction. We show that self-consistency is essential in $GW$-based calculations to obtain the full spectral information. The second-order exchange diagram (i.e., a vertex correction) is also crucial to obtain the good spectral description of the plasmon satellites. We corroborate these results by considering conventional equilibrium $GW$-based calculations for the pure jellium model. We find that with no second-order vertex correction, one cannot obtain the full set of plasmon side-band resonances. We also discuss in detail the formal expression of the Dyson equations obtained for the time-ordered Green's function at zero and finite temperature from the Keldysh formalism and from conventional equilibrium many-body perturbation theory.