A characterization of the Anderson metal-insulator transport transition

Abstract

We investigate the Anderson metal-insulator transition for random Schrödinger operators. We define the strong insulator region to be the part of the spectrum where the random operator exhibits strong dynamical localization in the Hilbert-Schmidt norm. We introduce a local transport exponent β(E), and set the metallic transport region to be the part of the spectrum with nontrivial transport (i.e., β(E) > 0). We prove that these insulator and metallic regions are complementary sets in the spectrum of the random operator, and that the local transport exponent β(E) provides a characterization of the metal-insulator transport transition. Moreover, we show that if there is such a transition, then β(E) has to be discontinuous at a transport mobility edge. More precisely, we show that if the transport is nontrivial then β(E) ≥ 1 2d , where d is the space dimension. These results follow from a proof that slow time evolution of quantum waves in random media implies the starting hypothesis for the authors' bootstrap multiscale analysis. We also conclude that the strong insulator region coincides with the part of the spectrum where we can perform a bootstrap multiscale analysis, proving that the multiscale analysis is valid all the way up to a transport mobility edge. Contents

Explicit finite volume criteria for localization in continuous random media and applications

“…In [GK3] we defined the strong insulator region Σ SI for H ω by [GK3,Theorem 4.2]), and we must have nontrivial transport in Σ I \Σ I MSA . In this article we provide three explicit finite volume criteria for localization in Theorems 2.4, 2.5, and 2.6.…”

“…[CoH1], [K], [St], [GK3]). In particular, [GK3,Theorem A.1] proves Asumption SLI with the constant γ I in (2.7) given by…”

“…Assumption SLI inequality can be proven as in [GK3,Theorem A.1] for any open interval I, with the constant γ I in (2.7) given by (4.18) where the constantγ depends only on M ; it does not depend on B. (Although, as noted in [CoH2], we have…”

“…These criteria thus yield Anderson localization, strong dynamical localization, SULE (semiuniformly localized eigenfunctions), etc. (See also [GK3,5].) These explicit finite volume criteria are particularly useful in situations where the crucial quantities of the model that enter the multiscale analysis (the constant in Wegner's estimate and the constant in the SimonLieb inequality) depend on the parameters of the model (e.g.…”

Linear response theory for magnetic Schrödinger operators in disordered media

“…The main challenge is to allow for the Fermi energy E F to be inside a region of localization, as described for random operators in [AENSS,AG,GK1,GK3]. Note that the existence of these regions of localization has been proven for random Landau Hamiltonians with Anderson-type potentials [CH,GK4,W], and that assumption (1.4) holds in these regions of localization [BoGK,GK5].…”