The Small Polaron Crossover: Role of Dimensionality

Abstract

The crossover from quasi free electron to small polaron in the Holstein model for a single electron is studied by means of both exact and self-consistent calculations in one dimension and on an infinite coordination lattice, in order to understand the role of dimensionality in such a crossover. We show that a small polaron ground-state occurs when both strong coupling ($λ>1$) and multiphonon ($α^2 >1$) conditions are fulfilled leading to different relevant coupling constants ($λ$) in adiabatic and ($α^2$) anti adiabatic region of the parameters space. We also show that the self-consistent calculations obtained by including the first electron-phonon vertex correction give accurate results in a sizeable region of the phase diagram well separated from the polaronic crossover.