The small-polaron crossover: Comparison between exact results and vertex correction approximation

We study the crossover from quasi-free electron to small polaron in the Holstein model for a single electron by means of both exact and self-consistent calculations in one dimension and on an infinite coordination lattice. We show that the crossover occurs when both strong coupling (lambda > 1) and multiphonon (alpha(2) > 1) conditions are fulfilled leading to different relevant coupling constants (lambda) in the adiabatic and (alpha(2)) antiadiabatic 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.