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

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.

29 pages, no figures

References in corpus (12)

• Charge Deficiency, Charge Transport and Comparison of Dimensions
• Simultaneous quantization of edge and bulk Hall conductivity
• Equality of bulk and edge Hall conductance revisited
• On the extended nature of edge states of Quantum Hall Hamiltonians
• On edge states in semi-infinite quantum Hall systems
• Characterization of the Spectrum of the Landau Hamiltonian with Delta Impurities
• Propagating edge states for a magnetic Hamiltonian
• Lifetimes of impurity states in crossed magnetic and electric fields
• Extended Edge States in Finite Hall Systems
• Spectral cross correlations of magnetic edge states
• Spectral Properties of Finite Quantum Hall Systems
• Spectral flow and level spacing of edge states for quantum Hall hamiltonians

Cited by in corpus (8)

• Spectra of soft ring graphs
• Persistent currents for 2D Schrödinger operator with a strong $\delta$-interaction on a loop
• Edge Currents for Quantum Hall Systems, I. One-Edge, Unbounded Geometries
• Quantization of edge currents along magnetic barriers and magnetic guides
• Resonances Width in Crossed Electric and Magnetic Fields
• Extended Edge States in Finite Hall Systems
• Spectral Properties of Finite Quantum Hall Systems
• Spectral flow and level spacing of edge states for quantum Hall hamiltonians