On the extended nature of edge states of Quantum Hall Hamiltonians

TL;DR

This paper investigates the extended nature of edge states in Quantum Hall systems, demonstrating their robustness against disorder and applying Mourre theory to prove spectral properties.

Contribution

It provides a rigorous analysis of edge state extension and spectral continuity in Quantum Hall Hamiltonians, especially under disorder and boundary conditions.

Findings

Edge states remain extended with non-zero edge current despite disorder.

Spectral analysis shows absolute continuity between Landau levels.

Mourre theory confirms the extended nature of edge states.

Abstract

Properties of eigenstates of one-particle Quantum Hall Hamiltonians localized near the boundary of a two-dimensional electron gas - so-called edge states - are studied. For finite samples it is shown that edge states with energy in an appropriate range between Landau levels remain extended along the boundary in the presence of a small amount of disorder, in the sense that they carry a non-zero chiral edge current. For a two-dimensional electron gas confined to a half-plane, the Mourre theory of positive commutators is applied to prove absolute continuity of the energy spectrum well in between Landau levels, corresponding to edge states.