Generalized Bethe Ansatz Equations for Hofstadter Problem

TL;DR

This paper derives generalized Bethe ansatz equations for the Hofstadter problem, describing electrons on 2D lattices in magnetic fields, extending previous results to high genus algebraic curves.

Contribution

It introduces a unified framework of Bethe ansatz equations for the Hofstadter problem on various lattice geometries and flux conditions, generalizing existing solutions.

Findings

Derived Bethe ansatz equations for high genus algebraic curves.

Connected new formulas to previous results in the trigonometric case.

Provided a mathematical structure for analyzing electron dynamics in magnetic fields.

Abstract

The problem of diagonalization of the quantum mechanical Hamiltonian, governing dynamics of an electron on a two-dimensional triangular or square lattice in external uniform magnetic field, applied perpendicularly to the lattice plane, the flux through lattice cell, divided by the elementary quantum flux, being rational number, is reduced to generalized Bethe ansatz like equations on high genus algebraic curve. Our formulae for the trigonometric case, where genus of the curve vanishes, contain as a particular case recent result of Wiegmann and Zabrodin.