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

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.