Localisation for non-monotone Schrödinger operators

Abstract

We study localisation effects of strong disorder on the spectral and dynamical properties of (matrix and scalar) Schrödinger operators with nonmonotone random potentials, on the d-dimensional lattice. Our results include dynamical localisation, i.e. exponentially decaying bounds on the transition amplitude in the mean. They are derived through the study of fractional moments of the resolvent, which are finite due to resonance-diffusing effects of the disorder. One of the byproducts of the analysis is a nearly optimal Wegner estimate. A particular example of the class of systems covered by our results is the discrete alloy-type Anderson model.where the entries v(n) of the potential are random and independent.The basic phenomenon, named Anderson localisation after the physicist P. W. Anderson, is that disorder can cause localisation of electron states, which manifests itself in time evolution (non-spreading of wave packets), (vanishing of) conductivity in response to electric field, Hall currents in the presence of both magnetic and electric field, and statistics of the spacing between nearby energy levels. The first property implies spectral localisation, i.e. the spectral measure of H A is almost surely pure point, and almost sure exponential decay of eigenfunctions.These properties are known to hold for H A in each of the following cases: 1) high disorder (the coupling constant g is large), 2) extreme energies, 3) weak disorder away from the spectrum of the unperturbed operator, and 4) one dimension, d = 1.Historically, the first proof of spectral localisation was given by Goldsheid, Molchanov and Pastur [12], for a one-dimensional continuous random Schrödinger operator.In higher dimension, the absence of diffusion was first established in 1983 by Fröhlich and Spencer [9] using multi-scale analysis. Their approach has led to a multitude of results on localisation for a wide range of problems. The reader is referred to the monograph of Stollmann [17] or the recent lecture notes of W. Kirsch [14] for a review of the history of the subject and a gentle introduction to the multiscale analysis -which is not used here.One of the ingredients of multi-scale analysis is the regularity of the integrated density of states, the (distribution function of the) average of the spectral measure over the randomness.Ten years later Aizenman and Molchanov [2] introduced an alternative method for the proof of localisation, known as the fractional moment method, which has also found numerous applications. In particular, in [1], Aizenman introduced the notion of eigenfunction correlator, which, combined with the fractional moment method, allowed him to give the first proof of dynamical localisation. We refer to the lecture notes of Stolz [19] and Aizenman and Warzel [3] for a survey of subsequent developments.In the fractional moment method, an a priori estimate on the diagonal elements of the resolvent (H A − λ) −1 plays a key rôle in the underlying analysis.

Eigenvalue counting inequalities, with applications to Schrödinger operators

“…Using this result in conjunction with (3.6) we obtain the there exist ǫ 0 > 0 and b > 0 that depend on k, m, α so that 8) for N ≤ m and for ǫ < ǫ 0 . The combination of Lemma 3.3 and (3.8) yields (1.9).…”

“…The Wegner estimate for a random block operator H block -with V (x) = gv(x)A where v(x) are independent random variables and A is a fixed invertible Hermitian matrix-holds in perturbative regimes. That is, it holds near the edges of the spectrum, [4] and in the strong disorder regime 1 ≪ g, [8]. A weaker bound (weaker in terms of the volume dependence) near the edges of the spectrum was established for Fröhlich model, where the matrix-valued potential is given by…”

Localization for Random Block Operators Related to the XY Spin Chain

“…The papers [14] and [15] have provided localization proofs for random block operators. The authors of [15], building on [22], use a Wegner estimate and a Lifshitz tail bound and adapt the bootstrap multiscale analysis of Germinet and Klein to prove spectral and dynamical localization at internal band edges for a class of random block operators.…”

“…However, random block operators have recently found considerable interest, see [22,14,15,13], with physical motivation also provided by the Bogoliubov-de Gennes equation in the mean-field approximation of BCS theory. We also mention that the model given by the anisotropic XY chain has been considered in the quantum information literature under the name Majorana chain, see [23] and [6].…”

“…The authors of [15], building on [22], use a Wegner estimate and a Lifshitz tail bound and adapt the bootstrap multiscale analysis of Germinet and Klein to prove spectral and dynamical localization at internal band edges for a class of random block operators. In [14] the fractional moment method is adapted to prove localization of the entire spectrum for a large class of random block operators (including those arising from the anisotropic XY chain) in the large disorder regime.…”