Extended Edge States in Finite Hall Systems

TL;DR

This paper proves that in finite quantum Hall systems, edge states with well-defined positive or negative velocities exist in the spectral gap, extending understanding of edge states to systems with two boundaries.

Contribution

It establishes the existence and properties of edge states in finite Hall systems with two boundaries, generalizing previous single-boundary results.

Findings

Edge states have strictly positive or negative average velocities.

Spectrum consists of two sets of eigenenergies associated with edge states.

Results hold with high probability for large system sizes.

Abstract

We study edge states of a random Schroedinger operator for an electron submitted to a magnetic field in a finite macroscopic two dimensional system of linear dimensions equal to L. The y direction is L-periodic and in the x direction the electron is confined by two smoothly increasing parallel boundary potentials. We prove that, with large probability, for an energy range in the first spectral gap of the bulk Hamiltonian, the spectrum of the full Hamiltonian consists only on two sets of eigenenergies whose eigenfuntions have average velocities which are strictly positive/negative, uniformly with respect to the size of the system. Our result gives a well defined meaning to the notion of edge states for a finite cylinder with two boundaries, and extends previous studies on systems with only one boundary.