Quantization of the Hall conductance and delocalization in ergodic Landau Hamiltonians

TL;DR

This paper proves the quantization and continuity of the Hall conductance in ergodic Landau Hamiltonians and demonstrates the existence of delocalized states near Landau levels, indicating a phase transition in these quantum systems.

Contribution

It establishes the quantization of Hall conductance under decay conditions and proves delocalization near Landau levels, advancing understanding of quantum Hall systems.

Findings

Quantization of Hall conductance under decay conditions.

Existence of delocalized states near Landau levels.

Transition between localization and delocalization in Landau bands.

Abstract

We prove quantization of the Hall conductance for continuous ergodic Landau Hamiltonians under a condition on the decay of the Fermi projections. This condition and continuity of the integrated density of states are shown to imply continuity of the Hall conductance. In addition, we prove the existence of delocalization near each Landau level for these two-dimensional Hamiltonians. More precisely, we prove that for some ergodic Landau Hamiltonians there exists an energy $E$ near each Landau level where a ``localization length'' diverges. For the Anderson-Landau Hamiltonian we also obtain a transition between dynamical localization and dynamical delocalization in the Landau bands, with a minimal rate of transport, even in cases when the spectral gaps are closed.