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

Abstract

We study continuous Anderson Hamiltonians with non-degenerate single site probability distribution of bounded support, without any regularity condition on the single site probability distribution. We prove the existence of a strong form of localization at the bottom of the spectrum, which includes Anderson localization (pure point spectrum with exponentially decaying eigenfunctions) with finite multiplicity of eigenvalues, dynamical localization (no spreading of wave packets under the time evolution), decay of eigenfunctions correlations, and decay of the Fermi projections. We also prove log-Hölder continuity of the integrated density of states at the bottom of the spectrum.

Many-body localization in the droplet spectrum of the random XXZ quantum spin chain

“…(i, j; I) is the relevant eigenfunction correlator for H (L) N corresponding to a finite interval I ⊂ R and a pair of indices i, j ∈ Z. Our proof follows the outlines of the proofs of dynamical localization for random Schrödinger operators given in[21,22,23,32]. The key ingredients are Lemma 7.2 and the a priori bound on eigenfunction correlators derived in Lemma 8.2 below.For a given N ∈ N, let P 1 be the indicator function of X(L) N,1 := X N,1 ∩ X Note that, due to working in finite volume, all spectra are finite, and thatĤ (L) N has no spectrum below 2(1 − 1 ∆ ).…”

Manifestations of Dynamical Localization in the Disordered XXZ Spin Chain

“…The basic phenomenon of Anderson localization in the single particle framework is that disorder can cause localization of electron states and thereby manifest itself in properties such as non-spreading of wave packets under time evolution and absence of dc transport. The mechanism behind this behavior is well understood by now, both physically and mathematically (e.g., [6,18,28,22,4,15]). Many manifestations of single-particle Anderson localization remain valid if one considers a fixed number of interacting particles, e.g., [13,3,29].…”

Chern Numbers, Localisation and the Bulk-edge Correspondence for Continuous Models of Topological Phases

“…We return to our initial motivation in Section 7 and include a case-study of how our theory applies to disordered quantum systems and their topological properties. The example of disordered magnetic Schrödinger operators on L 2 (R d ) also allows us to consider the connection of our Sobolev algebra to the localised states studied in [1,36,37]. We compare our results and those in [1], where we show that if the Fermi energy lies in a region of dynamical localisation and the disorder space has an ergodic probability measure, then our Z or Z 2 -valued bulk indices are still well-defined.…”