Abstract:
Electron-phonon interactions (EPIs) are ubiquitous in condensed matter physics and materials science, and are crucial to understanding many phenomena including polarons. In this defense, we present several advancements in approximate methods for polaronic problems. We present a study of the accuracy of the second- and fourth-order cumulant expansions (CE) of the electronic Green's function, and find that the second-order cumulant expansion can be a useful tool for determining spectral functions while the fourth-order CE introduces pathologies that may persist at arbitrarily high-order. We introduce a new self-consistent cumulant expansion (SC-CE) which remedies many of the deficits of the CE, and can produce accurate spectra across the entire Brillouin-zone, but the trade-off for this increased accuracy is the introduction of negative spectral-weight and the potential for rapid divergences in time. These problems can be minimized in the thermodynamic limit and in more realistic cases where phonon dispersion exists. We finally consider a new method to determine the ground-state structure of the polaron in ab initio materials. We present a new all-coupling variational method, and consider the ab initio case of lithium fluoride (LiF). We find that our ansatz provides accurate polaron binding energies for both the hole-polaron and the electron-polaron which are classical cases of small and large polarons, respectively. Future developments and applications of the methodologies presented in this defense may enable quantitative calculations of polarons in large-intermediately coupled ab initio systems, such as the lead-halide perovskites and other systems where it has hitherto been difficult to fully understand the effects of the electron-phonon interactions.
