Anomalous propagators and the particle-particle channel: Hedin's equations

TL;DR

This paper extends Hedin's equations to include anomalous propagators and pairing potentials, deriving a new self-consistent framework that incorporates the particle-particle T-matrix approximation and allows systematic improvements.

Contribution

It introduces a novel set of self-consistent equations extending Hedin's framework to the particle-particle channel with vertex corrections.

Findings

Derivation of a new self-consistent set of equations for anomalous propagators.

Equivalence to Hedin's equations with a first-order particle-particle T-matrix approximation.

Framework enables systematic inclusion of vertex corrections beyond T-matrix.

Abstract

Hedin's equations provide an elegant route to compute the exact one-body Green's function (or propagator) via the self-consistent iteration of a set of non-linear equations. Its first-order approximation, known as $GW$, corresponds to a resummation of ring diagrams and has shown to be extremely successful in physics and chemistry. Systematic improvement is possible, although challenging, via the introduction of vertex corrections. Considering anomalous propagators and an external pairing potential, we derive a new self-consistent set of closed equations equivalent to the famous Hedin equations but having as a first-order approximation the particle-particle (pp) $T$-matrix approximation where one performs a resummation of the ladder diagrams. This pp version of Hedin's equations offers a way to go systematically beyond the $T$-matrix approximation by accounting for low-order pp vertex corrections.