The Small Polaron Crossover: Role of Dimensionality

TL;DR

This paper investigates how the transition from free electrons to small polarons in the Holstein model depends on dimensionality, using exact and self-consistent methods across different lattice structures.

Contribution

It provides a detailed analysis of the conditions for small polaron formation and evaluates the accuracy of self-consistent calculations with vertex corrections.

Findings

Small polarons form under strong coupling and multiphonon conditions.

Self-consistent calculations with vertex corrections are accurate in certain parameter regions.

Dimensionality significantly influences the polaron crossover behavior.

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 ($\lambda>1$) and multiphonon ($\alpha^2 >1$) conditions are fulfilled leading to different relevant coupling constants ($\lambda$) in adiabatic and ($\alpha^2$) 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.