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

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.

68 pages

References in corpus (15)

• The Non-Commutative Geometry of the Quantum Hall Effect
• Simultaneous quantization of edge and bulk Hall conductivity
• Equality of bulk and edge Hall conductance revisited
• Equality of the bulk and edge Hall conductances in a mobility gap
• On the extended nature of edge states of Quantum Hall Hamiltonians
• Boundary maps for $C^{\ast }$-crossed products with R with an application to the quantum Hall effect
• On edge states in semi-infinite quantum Hall systems
• Characterization of the Spectrum of the Landau Hamiltonian with Delta Impurities
• Edge currents in the absence of edges
• Intermixture of extended edge and localized bulk energy levels in macroscopic Hall systems
• Propagating edge states for a magnetic Hamiltonian
• Extended Edge States in Finite Hall Systems
• Magnetic transport in a straight parabolic channel
• Spectral Properties of Finite Quantum Hall Systems
• Scattering of magnetic edge states

Cited by in corpus (8)

• Absolutely Continuous Edge Spectrum of Hall Insulators on the Lattice
• Absolutely continuous edge spectrum of topological insulators with an odd time-reversal symmetry
• A geometric Iwatsuka type effect in quantum layers
• Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators
• Magnetic quantum currents in the presence of a Neumann wall
• Characterization of bulk states in one-edge quantum Hall systems
• On the ground state of the Laplacian in presence of a magnetic field created by a rectilinear current
• Magnetic transport in laterally coupled layers