  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.
