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

TL;DR

This paper extends Hedin's GW equations to the Keldysh contour, applying it to model systems at equilibrium, and demonstrates the importance of self-consistency and vertex corrections for accurate spectral features.

Contribution

The paper develops a formal extension of GW and vertex corrections on the Keldysh contour, with applications to model systems at equilibrium, highlighting the role of self-consistency and vertex corrections.

Findings

Self-consistency is crucial for accurate spectral information.

Vertex corrections improve the description of plasmon peaks.

Non-self-consistent calculations can reproduce vertex effects effectively.

Abstract

We provide the formal extension of Hedin's GW equations for single-particle Green's functions with electron-electron interaction onto the Keldysh time-loop contour. 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 diagrammatic perturbation expansion of the core-electron/plasmon coupling on the spectral functions of the so-called S-model which provides an exact solution, concentrating especially on the effects of self-consistency and vertex corrections on the GW self-energy. For the S-model, self-consistency is essential for GW-like calculations to obtain the full spectral information. The second- order exchange diagram (i.e. a vertex correction) is crucial to obtain a better spectral description of the plasmon peak and side-band peaks in comparison to GW-like calculations. However, the vertex corrections are well reproduced within a non-self-consistent calculation. We also consider conventional equilibrium GW calculations for the pure jellium model. We find that with no second-order vertex correction, we cannot obtain the full set of plasmon side-band peaks. Finally, we address the issues of formal connection for the Dyson equations of the time-ordered Green's function and the Keldysh Green's functions at equilibrium in the cases of zero and finite temperature.