Variational ground states of the Holstein Hamiltonian

The ground states of the Holstein Hamiltonian are studied via multi-variable variational calculations. Three Ansatze of incremental sophistication, namely, the small polaron Ansatz, Toyozawa's Ansatz, and a generalization of Toyozawa's Ansatz, have been investigated. By comparing the phase diagrams of the three Ansatze, it has been shown that the much debated discontinuity marking the crossover between the small and large polaron can be significantly reduced by increased complexity in the Ansatze. A numerically exact solution has been obtained for Toyozawa's Ansatz without further simplifications, and a unified phase diagram has been presented. Toyozawa's Ansatz reveals that below the self-trapping transition, the phonon part of the ground state wavefunction smoothly evolves with decreasing exciton-phonon coupling into two components: a one-phonon state of a wavevector close to the crystal wavevector, and a localized phonon structure which resembles that of the large polaron in the adiabatic limit. The more sophisticated Ansatz yields a significant lowering of variational energies and a narrowing of energy bandwidths with respect to Toyozawa's Ansatz in most parameter regimes. Nonlocal exciton-phonon coupling has firstly been investigated following the Munn-Silbey approach. Canonical transformation coefficients are numerically found to differ significantly from those based on analytic approximations. This has immediate implications in transport calculations. Variational methods are applied to test the validity of the Munn-Silbey approach. The ground state energy bands have been obtained for arbitrary transfer integrals, and local and nonlocal exciton-phonon coupling strength. In the absence of the transfer integral, Toyozawa's Ansatz and the Munn-Silbey approach are found to yield similar polaron structures and energy bands at zero temperature. The effect of the transfer integral, however, is underestimated in the Munn-Silbey approach. For zero local coupling, two types of discontinuous crossovers are found: one at the Brillouin zone center which resembles that of the local coupling only case, and another at the zone boundary which represents a crossover from a localized phonon structure to a structure that is predominantly a one-phonon plane wave. The latter crossover occurs smoothly in the local coupling only case. The variational approaches developed here can readily be generalized to finite temperatures to model real materials.