Many-body Green's function theory for electron-phonon interactions: the Kadanoff-Baym approach to spectral properties of the Holstein dimer

TL;DR

This paper develops a Kadanoff-Baym formalism to analyze time-dependent electron-phonon interactions in a Holstein dimer, revealing stability issues in certain approximations and limitations in describing spectral responses during bipolaronic crossover.

Contribution

It extends previous ground state studies by applying a time-dependent Green's function approach to spectral properties, highlighting stability conditions and response function limitations in different approximations.

Findings

Unstable dynamics occur with homogeneous ground states at strong interactions.

Fully self-consistent Born approximation avoids certain instabilities.

None of the approximations accurately describe the response during bipolaronic crossover.

Abstract

We present a Kadanoff-Baym formalism to study time-dependent phenomena for systems of interacting electrons and phonons in the framework of many-body perturbation theory. The formalism takes correctly into account effects of the initial preparation of an equilibrium state, and allows for an explicit time-dependence of both the electronic and phononic degrees of freedom. The method is applied to investigate the charge neutral and non-neutral excitation spectra of a homogeneous, two-site, two-electron Holstein model. This is an extension of a previous study of the ground state properties in the Hartree (H), partially self-consistent Born (Gd) and fully self-consistent Born (GD) approximations published in Ref. [arXiv:1403.2968]. We show that choosing a homogeneous ground state solution leads to unstable dynamics for a sufficiently strong interaction, and that allowing a symmetry-broken state prevents this. The instability is caused by the bifurcation of the ground state and understood physically to be connected with the bipolaronic crossover of the exact system. This mean-field instability persists in the partially self-consistent Born approximation but is not found for the fully self-consistent Born approximation. By understanding the stability properties, we are able to study the linear response regime by calculating the density-density response function by time-propagation. This functions amounts to a solution of the Bethe-Salpeter equation with a sophisticated kernel. The results indicate that none of the approximations is able to describe the response function during or beyond the bipolaronic crossover for the parameters investigated. Overall, we provide an extensive discussion on when the approximations are valid, and how they fail to describe the studied exact properties of the chosen model system.