Dynamical Localization for Unitary Anderson Models

TL;DR

This paper proves dynamical localization for certain unitary Anderson models on lattices, demonstrating exponential decay of Green function moments and spectral localization across various regimes and dimensions.

Contribution

It extends the method of Aizenman-Molchanov to unitary operators, establishing localization in regimes analogous to the self-adjoint case.

Findings

Exponential decay of fractional moments of the Green function in specified regimes.

Dynamical localization implies spectral localization for these models.

Results hold in multiple dimensions and disorder regimes.

Abstract

This paper establishes dynamical localization properties of certain families of unitary random operators on the d-dimensional lattice in various regimes. These operators are generalizations of one-dimensional physical models of quantum transport and draw their name from the analogy with the discrete Anderson model of solid state physics. They consist in a product of a deterministic unitary operator and a random unitary operator. The deterministic operator has a band structure, is absolutely continuous and plays the role of the discrete Laplacian. The random operator is diagonal with elements given by i.i.d. random phases distributed according to some absolutely continuous measure and plays the role of the random potential. In dimension one, these operators belong to the family of CMV-matrices in the theory of orthogonal polynomials on the unit circle. We implement the method of Aizenman-Molchanov to prove exponential decay of the fractional moments of the Green function for the unitary Anderson model in the following three regimes: In any dimension, throughout the spectrum at large disorder and near the band edges at arbitrary disorder and, in dimension one, throughout the spectrum at arbitrary disorder. We also prove that exponential decay of fractional moments of the Green function implies dynamical localization, which in turn implies spectral localization. These results complete the analogy with the self-adjoint case where dynamical localization is known to be true in the same three regimes.