Upper and lower bounds for the large polaron dispersion in $D=1,2,3$ dimensions

TL;DR

This paper provides numerical bounds and a simple fitting formula for the polaron dispersion in 1D, 2D, and 3D, demonstrating high accuracy and applicability across different materials and coupling strengths.

Contribution

It introduces a method to calculate tight bounds for polaron dispersion in various dimensions and proposes a universal fitting formula validated by comparisons.

Findings

Bounds are close for small electron-phonon couplings.

The fitting formula accurately describes dispersion at intermediate coupling.

Results are reliable, reproducible, and applicable across different materials.

Abstract

Numerical results for the polaron dispersion are presented for an arbitrary number of space dimensions. Upper and lower bounds are calculated for the dispersion curves. They are rather close to each other in the cases of small electron-phonon couplings usual for real polar materials. To describe the dispersion in other materials, we suggest a simple fitting formula which can be applied at intermediate values of the Fr\"ohlich electron-phonon coupling constant. Its validity is approved by the comparison with direct calculations and previously obtained results. This makes our results not only reliable and highly accurate but also easy reproducible.