An extended q-deformed su(2) algebra and the Bloch electron problem

TL;DR

This paper introduces an extended q-deformed su(2) algebra to model Bloch electrons in magnetic fields with periodic potentials, generalizing previous work and deriving Bethe ansatz equations with solutions linked to Askey Wilson polynomials.

Contribution

It presents a novel extended q-deformed algebra framework for Bloch electrons, connecting algebraic structures to physical solutions and special functions.

Findings

Derived functional Bethe ansatz equations for the system

Zero energy solutions expressed via Askey Wilson polynomials

Generalized previous models of Bloch electrons in magnetic fields

Abstract

It is shown that an extended q-deformed $su(2)$ algebra with an extra (``Schwinger '') term can describe Bloch electrons in a uniform magnetic field with an additional periodic potential. This is a generalization of the analysis of Bloch electrons by Wiegmann and Zabrodin. By using a representation theory of this q-deformed algebra, we obtain functional Bethe ansatz equations whose solutions should be functions of finite degree. It is also shown that the zero energy solution is expressed in terms of an Askey Wilson polynomial.