Edge Currents for Quantum Hall Systems, I. One-Edge, Unbounded Geometries

TL;DR

This paper establishes explicit lower bounds on edge currents in quantum Hall systems with one-edge unbounded geometries, demonstrating their stability under potential perturbations and linking them to the spectrum of the Hamiltonian.

Contribution

It provides the first rigorous bounds on edge currents for unbounded one-edge geometries and analyzes their stability under various potential perturbations.

Findings

Edge currents are localized between Landau levels.

Currents remain stable under small potential perturbations.

Hamiltonian exhibits intervals of absolutely continuous spectrum.

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 unbounded subsets of the plane by confining potential barriers. The edges of the confining potential barrier create edge currents. In this, the first of two papers, we prove explicit lower bounds on the edge currents associated with one-edge, unbounded geometries formed by various confining potentials. This work extends some known results that we review. The edge currents are carried by states with energy localized between any two Landau levels. These one-edge geometries describe the electron confined to certain unbounded regions in the plane obtained by deforming half-plane regions. We prove that the currents are stable under various potential perturbations, provided the perturbations are suitably small relative to the magnetic field strength, including perturbations by random potentials. For these cases of one-edge geometries, the existence of, and the estimates on, the edge currents imply that the corresponding Hamiltonian has intervals of absolutely continuous spectrum. In the second paper of this series, we consider the edge currents associated with two-edge geometries describing bounded, cylinder-like regions, and unbounded, strip-like, regions.