Delocalization for Random Landau Hamiltonians with Unbounded Random Variables

TL;DR

This paper proves a transition between localized and delocalized states in Landau Hamiltonians with unbounded random potentials, showing gaps close and localized states emerge, impacting quantum transport properties.

Contribution

It extends localization/delocalization analysis to unbounded random potentials and develops new Wegner estimates for such cases.

Findings

Gaps in Landau levels close with unbounded randomness

Localized states fill the gaps upon perturbation

A minimal transport rate is identified in the delocalized region

Abstract

In this note we prove the existence of a localization/delocalization transition for Landau Hamiltonians randomly perturbed by an electric potential with unbounded amplitude. In particular, with probability one, no Landau gaps survive as the random potential is turned on, the gaps close, filling up partly with localized states. A minimal rate of transport is exhibited in the region of delocalization. To do so, we exploit the a priori quantization of the Hall conductance and extend recent Wegner estimates to the case of unbounded random variables.