Cumulant methods for electron-phonon problems. II. The self-consistent cumulant expansion

TL;DR

This paper introduces a self-consistent cumulant expansion (SC-CE) method for electron-phonon problems, demonstrating its improved accuracy over standard perturbative approaches in the Holstein model, especially with phonon dispersion.

Contribution

The paper develops and tests a self-consistent cumulant expansion method, showing it corrects many issues of the standard perturbative cumulant expansion in electron-phonon models.

Findings

SC-CE improves accuracy over standard CE in the Holstein model.

Phonon dispersion stabilizes the SC-CE method.

Self-consistency corrects many perturbative CE failures, with some unphysical features.

Abstract

In this work we present a self-consistent cumulant expansion (SC-CE) and investigate its accuracy for the one-dimensional Holstein model with and without phonon dispersion. We show that for finite lattices sizes, the numerical integration of the SC-CE equations becomes unstable at long times. This defect is partially ameliorated when studying systems in the thermodynamic limit, enabling the demonstration that the SC-CE corrects many deficits of the standard perturbative CE in the (non-dispersive) Holstein model. The natural phonon damping that arises in the more realistic dispersed Holstein model renders the SC-CE stable, allowing for a complete assessment of the method. Here we find that self-consistency dramatically corrects many of the failures found in the perturbative CE, but also introduces some unphysical features. Finally, we comment on the potential use of SC-CE as a tool for calculating Green's functions in generic many-body problems.