Electron-phonon interactions (EPIs) are critically important for understanding many phenomena in solids, yet for many systems of interest exact methods are out of reach so approximate methods are necessary. Here, we consider the cumulant expansion and its ability to describe one-particle spectral information for the problem of a single electron coupled to phonons at finite-temperature. We examine the one-dimensional Holstein model with the standard cumulant expansion (CE) at second- and fourth-order, as well as a previously unexplored self-consistent cumulant expansion (SC-CE). For the second-order CE we find that at zero electronic momentum (k=0), the CE spectra accurately reproduces the main features of the VD spectrum across a range of temperatures. However, away from k=0 the second-order CE is only accurate at high temperatures. When it is well-behaved, the fourth-order makes some improvements over the second-order; however, it can display pathological behavior including divergence in time and non-physical negative spectral weight. The SC-CE is unstable at long times for small system sizes. Increasing the system size toward the thermodynamic limit does not entirely suppress the instabilities, but the SC-CE predicts many of the non-perturbative spectral features missed by the second-order CE at the expense of introducing some unphysical features. The Holstein model with dispersed phonons has natural damping present and stabilizes the SC-CE and we see dramatic improvement between the CE and SC-CE. It still predicts a small amount of non-physical negative spectral weight, but the success of the SC-CE at correctly predicting non-perturbative features of the spectra motivates our interest in applying the SC-CE to realistic systems.

Cumulant green's functions for electron-phonon problems: Perturbative and self-consistent expansions

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