Lattice Electrons on a Cylinder Surface in the Presence of Rational Magnetic Flux and Disorder

TL;DR

This paper investigates the quantum Hall effect in a disordered lattice electron system on a cylinder, analyzing spectral properties, localization, and Hall conductivity behavior under magnetic flux and disorder.

Contribution

It extends understanding of Hall conductivity and localization transitions in finite geometries with disorder and magnetic flux, connecting to the Diophantine equation.

Findings

Hall conductivity follows the Diophantine equation in mobility gaps.

Disorder causes band broadening and localized states in band tails.

Localization-delocalization transitions resemble those in the quantum Hall effect.

Abstract

We consider a disordered two-dimensional system of independent lattice electrons in a perpendicular magnetic field with rigid confinement in one direction and generalized periodic boundary conditions (GPBC) in the other direction. The objects investigated numerically are the orbits in the plane spanned by the energy eigenvalues and the corresponding center of mass coordinate in the confined direction, parameterized by the phase characterizing the GPBC. The Kubo Hall conductivity is expressed in terms of the winding numbers of these orbits. For vanishing disorder the spectrum of the system consists of Harper bands with energy levels corresponding to the edge states within the band gaps. Disorder leads to broadening of the bands. For sufficiently large systems localized states occur in the band tails. We find that within the mobility gaps of bulk states the Diophantine equation determines the value of the Hall conductivity as known for systems with torus geometry (PBCs in both directions). Within the spectral bands of extended states the Hall conductivity fluctuates strongly. For sufficiently large systems the generic behavior of localization-delocalization transitions characteristic for the quantum Hall effect are recovered.