Diagrammatic expansion for positive spectral functions beyond GW: Application to vertex corrections in the electron gas

TL;DR

This paper introduces a diagrammatic method to ensure positive spectral functions in many-body perturbation theory, specifically improving vertex corrections beyond GW for the electron gas, addressing negative spectral issues.

Contribution

The authors develop a novel diagrammatic approach that guarantees positive spectral functions and applies it to vertex corrections beyond GW in the electron gas.

Findings

Successfully constructs positive spectral functions with vertex corrections.

Derives minimal additional diagrams to fix GW spectral negativity.

Applies method to 3D electron gas with analytical and Monte Carlo techniques.

Abstract

We present a diagrammatic approach to construct self-energy approximations within many-body perturbation theory with positive spectral properties. The method cures the problem of negative spectral functions which arises from a straightforward inclusion of vertex diagrams beyond the GW approximation. Our approach consists of a two-steps procedure: we first express the approximate many-body self-energy as a product of half-diagrams and then identify the minimal number of half-diagrams to add in order to form a perfect square. The resulting self-energy is an unconventional sum of self-energy diagrams in which the internal lines of half a diagram are time-ordered Green's functions whereas those of the other half are anti-time-ordered Green's functions, and the lines joining the two halves are either lesser or greater Green's functions. The theory is developed using noninteracting Green's functions and subsequently extended to self-consistent Green's functions. Issues related to the conserving properties of diagrammatic approximations with positive spectral functions are also addressed. As a major application of the formalism we derive the minimal set of additional diagrams to make positive the spectral function of the GW approximation with lowest-order vertex corrections and screened interactions. The method is then applied to vertex corrections in the three-dimensional homogeneous electron gas by using a combination of analytical frequency integrations and numerical Monte-Carlo momentum integrations to evaluate the diagrams.