An SU(2) Analog of the Azbel--Hofstadter Hamiltonian

TL;DR

This paper introduces a modified Azbel--Hofstadter Hamiltonian modeled as a spin-S Euler top, revealing a solvable eigenvalue problem and spectral similarities with the original model, inspired by quantum symmetry considerations.

Contribution

It presents a new classical analog of the Azbel--Hofstadter Hamiltonian using a spin-S Euler top, connecting quantum symmetry with classical integrable systems.

Findings

Eigenvalue problem reduces to the S-gap Lamé equation.

Spectral shapes of the new model resemble those of the original for various spins.

Model is completely solvable in the coherent state representation.

Abstract

Motivated by recent findings due to Wiegmann and Zabrodin, Faddeev and Kashaev concerning the appearence of the quantum U_q(sl(2)) symmetry in the problem of a Bloch electron on a two-dimensional magnetic lattice, we introduce a modification of the tight binding Azbel--Hofstadter Hamiltonian that is a specific spin-S Euler top and can be considered as its ``classical'' analog. The eigenvalue problem for the proposed model, in the coherent state representation, is described by the S-gap Lam\'e equation and, thus, is completely solvable. We observe a striking similarity between the shapes of the spectra of the two models for various values of the spin S.