Dynamical delocalization in random Landau Hamiltonians

Abstract

We prove the existence of dynamical delocalization for random Landau Hamiltonians near each Landau level. Since typically there is dynamical localization at the edges of each disordered-broadened Landau band, this implies the existence of at least one dynamical mobility edge at each Landau band, namely a boundary point between the localization and delocalization regimes, which we prove to converge to the corresponding Landau level as either the magnetic field goes to infinity or the disorder goes to zero. In this article we prove the existence of dynamical delocalization for random Landau Hamiltonians near each Landau level. More precisely, we prove that for these two-dimensional Hamiltonians there exists at least one energy E near each Landau level such that ! (E) ! 1 , where ! (E), the local transport exponent introduced in [GK5], is a measure of the rate of transport in wave packets with spectral support near E. Since typically there is dynamical localization at the edges of each disordered-broadened Landau band, this implies the existence of at least one dynamical mobility edge at each Landau band, namely a boundary point between the localization and delocalization regimes, which we prove to converge to the corresponding Landau level as either the magnetic field goes to infinity or the disorder goes to zero. Random Landau Hamiltonians are the subject of intensive study due to their links with the integer quantum Hall e! ect [Kli], for which von Klitzing received the 1985 Nobel Prize in Physics. They describe an electron moving in a very thin flat conductor with impurities under the influence of a constant magnetic field perpendicular to the plane of the conductor, and play an important role in the understanding of the quantum Hall e! ect [L], [AoA], [T], [H], [NT], [Ku], [Be], [AvSS], [BeES]. Laughlin’s argument [L], as pointed out by Halperin [H], uses the assumption that under weak disorder and strong magnetic field the energy spectrum consists of bands of extended states separated