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

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.

7 pages, 2 figures, Revtex

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• Impurity bound states as detectors of topological band structures revisited
• Stability of zero energy Dirac touchings in the honeycomb Hofstadter problem
• Hofstadter problem in higher dimensions
• The quantum group, Harper equation and the structure of Bloch eigenstates on a honeycomb lattice