# Single-boson exchange decomposition of the vertex function

## TL;DR

This paper introduces a physically interpretable, computationally efficient decomposition of the two-particle vertex function in the Anderson impurity model, emphasizing single-boson exchange processes across interaction strengths.

## Contribution

It proposes a new vertex decomposition method based on boson exchange that avoids matrix inversion, simplifying calculations and providing physical insights.

## Key findings

- Single-boson exchange captures most vertex information at weak coupling.
- The decomposition remains effective at larger interactions, describing low-energy fluctuations.
- The method is computationally lighter than parquet decomposition.

## Abstract

We present a decomposition of the two-particle vertex function of the single-band Anderson impurity model which imparts a physical interpretation of the vertex in terms of the exchange of bosons of three flavors. We evaluate the various components of the vertex for an impurity model corresponding to the half-filled Hubbard model within dynamical mean-field theory. For small values of the interaction almost the entire information encoded in the vertex function corresponds to single-boson exchange processes, which can be represented in terms of the Hedin three-leg vertex and the screened interaction. Also for larger interaction, the single-boson exchange still captures scatterings between electrons and the dominant low-energy fluctuations and provides a unified description of the vertex asymptotics. The proposed decomposition of the vertex does not require the matrix inversion of the Bethe-Salpeter equation. Therefore, it represents a computationally lighter and hence more practical alternative to the parquet decomposition.