Mean-field theory for fermion pairs and the ab initio particle-vibration coupling approach

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.