Delocalization for random Landau Hamiltonians with unbounded random variables

Abstract

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.

A comprehensive proof of localization for continuous Anderson models with singular random potentials

“…In particular, there is no proof of localization for the multidimensional discrete Bernoulli-Anderson model, for which everything in [BoK] and this paper is valid except for the quantitative unique continuation principle; there is no unique continuation principle for discrete Schrödinger operators, where non-zero eigenfunctions may vanish on arbitrarily large sets [J,Theorem 2]. The same applies to random Landau Hamiltonians [CoH2,W1,GKS1,GKS2,GKM], where, although the unique continuation principle holds, an appropriate quantitative unique continuation principle is missing. (There is a quantitative unique continuation principle for Landau Hamiltonians, but it comes with the exponent 2 instead of 4 3 [Da].…”

Splitting of the Landau levels by magnetic perturbations and Anderson transition in 2D-random magnetic media

“…In this paper we consider 2D-random magnetic perturbations of the Landau Hamiltonian, and prove a transition between dynamical localization and dynamical delocalization inside an arbitrary number of bands. For our model, the phenomenon is thus similar to that arising for random electric potentials [GKS1,GKS2,GKM]. The proof of localization exploits the Wegner estimate of Hislop and Klopp [HK], revisited by Ghribi, Hislop and Klopp [GhHK], together with a simple weak disorder argument to start the multiscale analysis, provided some information on the location of the spectrum that we address in a separate argument.…”

“…for all η > 0, but as η gets small, the disorder becomes weaker in the sense that for most γ the coupling ω γ is small. We may speak of a diluted random model (see [GKS1,GKM] for a similar type of randomness). We denote by H B,λ,ω,η := H (A 0 + λA ω,η ), λ > 0, the corresponding magnetic random operator and will consider small values of the coupling constant λ.…”

“…In quantum Hall systems, namely a 2DEG submitted to a transverse constant magnetic field, localized states are responsible for the celebrated plateaux of the quantum Hall effect. In the case where the Hall conductance is discontinuous, non-trivial transport has been proved to take place in [GKS1] for electric disorder (see also [BeES,GKS2,GKM]). In this paper, we provide a similar picture but with magnetic disorder.…”

Mathematics of magic angles in a model of twisted bilayer graphene

“…If the disorder is signed, then classical Wegner-type estimates rule out the presence of flat bands under disorder, since the integrated density of states does not exhibit any jump discontinuities. This has been implemented for magnetic Schrödinger operators [13,20,21,22] and for twisted bilayer graphene by the authors in [5].…”