Edge Currents for Quantum Hall Systems, II. Two-Edge, Bounded and Unbounded Geometries

TL;DR

This paper analyzes edge currents in quantum Hall systems with various geometries, providing explicit bounds and demonstrating their stability under perturbations, including randomness, regardless of spectral properties.

Contribution

It offers new explicit lower bounds on edge currents for two-edge geometries and proves their stability under small perturbations, extending previous results.

Findings

Edge currents are stable under small perturbations.

Explicit lower bounds on edge currents are established.

Edge currents exist regardless of spectral type.

Abstract

Devices exhibiting the integer quantum Hall effect can be modeled by one-electron Schroedinger operators describing the planar motion of an electron in a perpendicular, constant magnetic field, and under the influence of an electrostatic potential. The electron motion is confined to bounded or unbounded subsets of the plane by confining potential barriers. The edges of the confining potential barriers create edge currents. This is the second of two papers in which we review recent progress and prove explicit lower bounds on the edge currents associated with one- and two-edge geometries. In this paper, we study various unbounded and bounded, two-edge geometries with soft and hard confining potentials. These two-edge geometries describe the electron confined to unbounded regions in the plane, such as a strip, or to bounded regions, such as a finite length cylinder. We prove that the edge currents are stable under various perturbations, provided they are suitably small relative to the magnetic field strength, including perturbations by random potentials. The existence of, and the estimates on, the edge currents are independent of the spectral type of the operator.