Multiplicity of solutions to GW-type approximations

TL;DR

This paper demonstrates that GW-type approximation equations have many solutions, but current methods find the physically correct one, supported by theorems and an efficient algorithm for advanced vertex corrections.

Contribution

It proves the uniqueness of the physical solution in GW approximations and introduces a new algorithm for self-consistent vertex corrections beyond GW.

Findings

Current GW methods find the correct physical solution.

Theorems explain solution uniqueness in GW equations.

An efficient algorithm for advanced vertex corrections is validated.

Abstract

We show that the equations underlying the $GW$ approximation have a large number of solutions. This raises the question: which is the physical solution? We provide two theorems which explain why the methods currently in use do, in fact, find the correct solution. These theorems are general enough to cover a large class of similar algorithms. An efficient algorithm for including self-consistent vertex corrections well beyond $GW$ is also described and further used in numerical validation of the two theorems.