Applicability of the cumulant expansion method for the calculation of transport properties in electron-phonon systems

TL;DR

This paper evaluates the cumulant expansion method's accuracy for calculating charge mobility in electron-phonon systems, comparing it with other established approaches using simplified models.

Contribution

It demonstrates that the cumulant expansion within the independent-particle approximation is accurate for weak to moderate coupling and moderate temperatures in specific models.

Findings

CE within IPA yields accurate results at weak to moderate coupling.

Comparison with Boltzmann and Migdal approaches shows good agreement.

Vertex corrections are discussed for the Peierls model.

Abstract

We assess the accuracy of the cumulant expansion (CE) method, combined with the independent-particle approximation (IPA), for calculating charge mobility in electron-phonon systems. As representative testbeds, we consider the Peierls and Fr\"ohlich models, which serve as simplified frameworks where accurate or numerically exact benchmarks are available. These are used to compare the CE results with those obtained using the Boltzmann formalism, the Migdal approximation, and its self consistent extension-approaches that are presently the most commonly employed alternatives for transport calculations. Supported by analytical arguments based on spectral sum rules and by our previous results for the Holstein model, we argue that, for weak to moderate coupling strengths and not-too-low temperatures, the CE within the IPA framework yields accurate results. In the case of the Peierls model, the role of vertex corrections is also discussed.