Intermixture of extended edge and localized bulk energy levels in macroscopic Hall systems

TL;DR

This paper analyzes the spectral properties of a random Schrödinger operator in a finite 2D system under a magnetic field, revealing a mixture of extended edge states and localized bulk states relevant to the quantum Hall effect.

Contribution

It provides a detailed classification of eigenvalues based on their quantum current, highlighting the coexistence of extended and localized states in finite quantum Hall systems.

Findings

Existence of O(L) eigenvalues with significant current

Presence of O(L^2) eigenvalues with exponentially small current

Relevance to understanding the integer quantum Hall effect

Abstract

We study the spectrum of a random Schroedinger operator for an electron submitted to a magnetic field in a finite but macroscopic two dimensional system of linear dimensions equal to L. The y direction is periodic and in the x direction the electron is confined by two smooth increasing boundary potentials. The eigenvalues of the Hamiltonian are classified according to their associated quantum mechanical current in the y direction. Here we look at an interval of energies inside the first Landau band of the random operator for the infinite plane. In this energy interval, with large probability, there exist O(L) eigenvalues with positive or negative currents of O(1). Between each of these there exist O(L^2) eigenvalues with infinitesimal current O(exp(-cB(log L)^2)). We explain what is the relevance of this analysis to the integer quantum Hall effect.