# Mean-Field Theory for Fermion Pairs and the ab initio Particle-Vibration-Coupling Approach

## TL;DR

This paper develops a mean-field theory for fermion pairs using a Dyson Bethe-Salpeter equation with frequency-dependent kernels, coupled with a particle-vibration coupling approach, showing promising numerical results for model systems.

## Contribution

It introduces a coupled Dyson-BSE and particle-vibration coupling framework for fermion pairs, integrating static mean fields with dynamic correlation effects.

## Key findings

- Numerical results for Lipkin model are promising.
- Numerical results for 1D Hubbard chain are promising.
- The approach effectively captures particle-vibration interactions.

## Abstract

A Dyson Bethe-Salpeter equation (Dyson-BSE) for fermion pairs is presented whose kernel has a static and a one frequency dependent contribution, analogous to the self energy of the single particle Dyson equation with the (static) mean field term and the energy dependent correlation term. The static part of the Dyson-BSE is the self-consistent mean field for the vibrations. At the same time, for the correlated single particle self-energy a full particle-vibration coupling (PVC) scattering equation is established where the vibration is the same as obtained from the Dyson-BSE. Both equations, single particle Dyson equation and Dyson-BSE, are coupled through self-consistency. Numerical results for Lipkin and 1D Hubbard chain are very promising.