Bethe ansatz for the Harper equation: Solution for a small commensurability parameter

TL;DR

This paper applies the Bethe ansatz to analyze the Harper equation, deriving spectral boundaries for small commensurability parameters and confirming results with semiclassical methods.

Contribution

It provides a novel solution for the Harper equation using Bethe ansatz in the small alpha limit, linking root distributions to spectral boundaries.

Findings

Distribution of Bethe roots for small alpha

Calculated spectral boundaries match semiclassical predictions

Validated Bethe ansatz approach for incommensurate systems

Abstract

The Harper equation describes an electron on a 2D lattice in magnetic field and a particle on a 1D lattice in a periodic potential, in general, incommensurate with the lattice potential. We find the distribution of the roots of Bethe ansatz equations associated with the Harper equation in the limit as alpha=1/Q tends to 0, where alpha is the commensurability parameter (Q is integer). Using the knowledge of this distribution we calculate the higher and lower boundaries of the spectrum of the Harper equation for small alpha. The result is in agreement with the semiclassical argument, which can be used for small alpha.