Functional approach to the electronic and bosonic dynamics of many-body systems perturbed with an arbitrary strong electron-boson interaction

Abstract

We present a formal derivation of the many-body perturbation theory for a system of electrons and bosons subject to a nonlinear electron-boson coupling. The interaction is treated at an arbitrary high order of bosons scattered. The considered Hamiltonian includes the well-known linear coupling as a special limit. This is the case, for example, of the Holstein and Fr\"{o}hlich Hamiltonians. Indeed, whereas linear coupling have been extensively studied, the scattering processes of electrons with multiple bosonic quasiparticles are largely unexplored. We focus here on a self-consistent theory in terms of dressed propagators and generalize the Hedin's equations using the Schwinger technique of functional derivatives. The method leads to an exact derivation of the electronic and bosonic self-energies, expressed in terms of a new family of vertex functions, high order correlators and bosonic and electronic mean-field potentials. In the electronic case we prove that the mean-field potential is the $n$th-order extension of the well-known Debye-Waller potential. We also introduce a bosonic mean-field potential entirely dictated by nonlinear electron-boson effects. The present scheme, treating electrons and bosons on an equal footing, demonstrates the full symmetry of the problem. The vertex functions are shown to have purely electronic and bosonic character as well as a mixed electron-boson one. These four vertex functions are shown to satisfy a generalized Bethe-Salpeter equation. Multi bosons response functions are also studied and explicit expressions for the two and the three bosons case are given.