Cumulant expansion in the Holstein model: Spectral functions and mobility

TL;DR

This study evaluates the second-order cumulant expansion's accuracy for spectral functions and mobility in the Holstein model, comparing it with DMFT and other approximations across various parameters and dimensions.

Contribution

It introduces an efficient numerical implementation of CE, benchmarks it against DMFT, and analyzes its validity and accuracy in different regimes and dimensions.

Findings

CE accurately captures quasiparticle and satellite peaks at intermediate coupling in 1D.

Charge mobility follows power laws at high temperatures, with different exponents for weak and strong coupling.

CE outperforms SCMA at stronger coupling and is more accurate than one-shot Migdal approximation in certain regimes.

Abstract

We examine the range of validity of the second-order cumulant expansion (CE) for the calculation of spectral functions, quasiparticle properties, and mobility of the Holstein polaron. We devise an efficient numerical implementation that allows us to make comparisons in a broad interval of temperature, electron-phonon coupling, and phonon frequency. For a benchmark, we use the dynamical mean-field theory (DMFT) which gives, as we have recently shown, rather accurate spectral functions in the whole parameter space, even in low dimensions. We find that in one dimension, the CE resolves well both the quasiparticle and the first satellite peak in a regime of intermediate coupling. At high temperatures, the charge mobility assumes a power law $\mu\propto T^{-2}$ in the limit of weak coupling and $\mu\propto T^{-3/2}$ for stronger coupling. We find that, for stronger coupling, the CE gives slightly better results than the self-consistent Migdal approximation (SCMA), while the one-shot Migdal approximation is appropriate only for a very weak electron-phonon interaction. We also analyze the atomic limit and the spectral sum rules. We derive an analytical expression for the moments in CE and find that they are exact up to the fourth order, as opposed to the SCMA where they are exact to the third order. Finally, we analyze the results in higher dimensions.