Asymmetric Bethe-Salpeter equation for pairing and condensation

TL;DR

This paper reexamines the Martin-Schwinger hierarchy, deriving an asymmetric Bethe-Salpeter equation that improves the description of pairing and condensation phenomena by avoiding unphysical repeated collisions and providing consistent gap equations.

Contribution

The paper introduces an asymmetric Bethe-Salpeter equation derived from hierarchical correlations, surpassing the parquet approximation and ensuring physical consistency in fermionic and bosonic systems.

Findings

Derivation of an asymmetric Bethe-Salpeter equation avoiding unphysical repeated collisions.

Establishment of a self-consistent framework that yields gap equations for fermions.

Demonstration that the asymmetric form is superior to symmetric approaches for physical accuracy.

Abstract

The Martin-Schwinger hierarchy of correlations are reexamined and the three-particle correlations are investigated under various partial summations. Besides the known approximations of screened, ladder and maximally crossed diagrams the pair-pair correlations are considered. It is shown that the recently proposed asymmetric Bethe-Salpeter equation to avoid unphysical repeated collisions is derived as a result of the hierarchical dependencies of correlations. Exceeding the parquet approximation we show that an asymmetry appears in the selfconsistent propagators. This form is superior over the symmetric selfconsistent one since it provides the Nambu-Gorkov equations and gap equation for fermions and the Beliaev equations for bosons while from the symmetric form no gap equation results. The selfenergy diagrams which account for the subtraction of unphysical repeated collisions are derived from the pair-pair correlation in the three-particle Greenfunction. It is suggested to distinguish between two types of selfconsistency, the channel-dressed propagators and the completely dressed propagators, with the help of which the asymmetric expansion completes the Ward identity and is $\Phi$-derivable.