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In this work we present a self-consistent cumulant expansion (SC-CE) andinvestigate its accuracy for the one-dimensional Holstein model with andwithout phonon dispersion. We show that for finite lattices sizes, thenumerical integration of the SC-CE equations becomes unstable at long times.This defect is partially ameliorated when studying systems in the thermodynamiclimit, enabling the demonstration that the SC-CE corrects many deficits of thestandard perturbative CE in the (non-dispersive) Holstein model. The naturalphonon damping that arises in the more realistic dispersed Holstein modelrenders the SC-CE stable, allowing for a complete assessment of the method.Here we find that self-consistency dramatically corrects many of the failuresfound in the perturbative CE, but also introduces some unphysical features.Finally, we comment on the potential use of SC-CE as a tool for calculatingGreen's functions in generic many-body problems.

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