Hofstadter Problem on the Honeycomb and Triangular Lattices: Bethe Ansatz Solution

TL;DR

This paper solves the Hofstadter problem on honeycomb and triangular lattices using Bethe Ansatz, providing a new analytical approach to understanding the energy spectrum under magnetic flux.

Contribution

It introduces a Bethe Ansatz solution for the Hofstadter problem on honeycomb and triangular lattices, enabling analysis of irrational flux limits.

Findings

Factorization of the honeycomb lattice problem into two triangular lattices.

Derivation of nested Bethe equations for eigenstates and energy spectrum.

Application of Thermodynamic Bethe Ansatz to analyze irrational flux.

Abstract

We consider Bloch electrons on the honeycomb lattice under a uniform magnetic field with $2 \pi p/q$ flux per cell. It is shown that the problem factorizes to two triangular lattices. Treating magnetic translations as Heisenberg-Weyl group and by the use of its irreducible representation on the space of theta functions, we find a nested set of Bethe equations, which determine the eigenstates and energy spectrum. The Bethe equations have simple form which allows to consider them further in the limit $p, q \to \infty$ by the technique of Thermodynamic Bethe Ansatz and analyze Hofstadter problem for the irrational flux.