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

TL;DR

This paper investigates the transition from free electron to small polaron in the Holstein model, comparing exact solutions with vertex correction approximations to identify the accuracy and relevant conditions of the crossover.

Contribution

It provides a detailed comparison between exact and approximate methods for the polaron crossover, highlighting the effectiveness of vertex correction calculations in certain parameter regimes.

Findings

Crossover occurs under strong coupling and multiphonon conditions.

Vertex correction approximations are accurate in a significant phase diagram region.

Different coupling constants are relevant in adiabatic and anti-adiabatic regimes.

Abstract

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 and multiphonon conditions are fulfilled leading to different relevant coupling constants in adiabatic and 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.