Cumulant methods for electron-phonon problems. I. Perturbative expansions

TL;DR

This paper evaluates the effectiveness of cumulant expansion methods in capturing spectral properties of electron-phonon systems, specifically the Holstein model, highlighting their strengths and limitations at various temperatures and momenta.

Contribution

It provides a comprehensive comparison of second- and fourth-order cumulant expansions with exact methods for the Holstein model, revealing their accuracy and issues across different regimes.

Findings

Second-order CE accurately describes spectral functions at zero momentum.

Fourth-order CE improves short-time dynamics but can cause divergences and unphysical results.

CE's accuracy varies with temperature and momentum, with limitations at non-zero momentum.

Abstract

In this work we investigate the ability of the cumulant expansion (CE) to capture one-particle spectral information in electron-phonon coupled systems at both zero and finite temperatures. In particular, we present a comprehensive study of the second- and fourth-order CE for the one-dimensional Holstein model as compared with numerically exact methods. We investigate both finite sized systems as well as the approach to the thermodynamic limit, drawing distinctions and connections between the behavior of systems in and away from the thermodynamic limit that enable a greater understanding of the ability of the CE to capture real-frequency information across the full range of wave vectors. We find that for zero electronic momentum, the spectral function is well described by the second-order CE at low and high temperatures. However, for non-zero electronic momenta, the CE is only accurate at high temperature. We analyze the fourth-order cumulant, and find that while it improves the description of the short-time dynamics encoded in the one-particle Green's function, it can introduce divergences in the time domain as well as unphysical negative spectral weight in the spectral function. When well-behaved, the fourth-order CE does provide notable accurate corrections to the second-order CE. Finally, we use our results to comment on the use of the CE as a tool for calculating transport behavior in the realistic ab initio modeling of materials.