Quantum Group and Magnetic Translations. Bethe-Ansatz Solution for Azbel-Hofstadter Problem

TL;DR

This paper introduces a novel approach linking magnetic translation groups to quantum groups, enabling Bethe-Ansatz solutions for the Azbel-Hofstadter problem and explicit zero mode wave functions.

Contribution

It establishes a connection between magnetic translations and quantum groups, providing Bethe-Ansatz solutions and explicit wave functions for the Azbel-Hofstadter problem.

Findings

Expressed the mid-band spectrum using Bethe-Ansatz equations.

Derived explicit zero mode wave functions in terms of q-deformed polynomials.

Presented solutions for isotropic and certain quasiperiodic cases.

Abstract

We present a new approach to the problem of Bloch electrons in magnetic ( sometimes called Azbel-Hofstadter problem) field, by making explicit a natural relation between the group of magnetic translations and the quantum group $U_{q}(sl_2)$. The approach allows us to express the "mid" band spectrum of the model and the Bloch wave function as solutions of the Bethe-Ansatz equations typical for completely integrable quantum systems. The zero mode wave functions are found explicitly in terms of $q$-deformed classical orthogonal polynomials.In this paper we present solution for the isotropic problem. We also present a class of solvable quasiperiodic equations related to $U_{q}(sl_2)$.