Quantum Monte Carlo and variational approaches to the Holstein model

Based on the canonical Lang-Firsov transformation of the Hamiltonian we develop a very efficient quantum Monte Carlo algorithm for the Holstein model with one electron. Separation of the fermionic degrees of freedom by a reweighting of the probability distribution leads to a dramatic reduction in computational effort. A principal component representation of the phonon degrees of freedom allows to sample completely uncorrelated phonon configurations. The combination of these elements enables us to perform efficient simulations for a wide range of temperature, phonon frequency, and electron-phonon coupling on clusters large enough to avoid finite-size effects. The algorithm is tested in one dimension and the data are compared with exact-diagonalization results and with existing work. Moreover, the ideas presented here can also be applied to the many-electron case. In the one-electron case considered here, the physics of the Holstein model can be described by a simple variational approach.