Quantum Group, Bethe Ansatz and Bloch Electrons in a Magnetic Field

TL;DR

This paper explores the wave functions of Bloch electrons in a magnetic field using quantum groups and Bethe Ansatz, revealing analytical solutions and multifractal structures for irrational flux values.

Contribution

It introduces a novel approach combining quantum group algebra and Bethe Ansatz to analyze Bloch electrons in magnetic fields, including irrational flux cases.

Findings

Analytical and numerical solutions for wave functions.

Multifractal structure of solutions for irrational flux.

Detailed analysis of semi-classical flux case.

Abstract

The wave functions for two dimensional Bloch electrons in a uniform magnetic field at the mid-band points are studied with the help of the algebraic structure of the quantum group $U_q(sl_2)$. A linear combination of its generators gives the Hamiltonian. We obtain analytical and numerical solutions for the wave functions by solving the Bethe Ansatz equations, proposed by Wiegmann and Zabrodin on the basis of above observation. The semi-classical case with the flux per plaquette $\phi=1/Q$ is analyzed in detail, by exploring a structure of the Bethe Ansatz equations. We also reveal the multifractal structure of the Bethe Ansatz solutions and corresponding wave functions when $\phi$ is irrational, such as the golden or silver mean.