Almost Everything About the Unitary Almost Mathieu Operator

TL;DR

This paper introduces a unitary almost-Mathieu operator derived from a quantum walk in a magnetic field, analyzing its spectral properties and duality, and classifying spectral types across parameter regions using advanced cocycle theory.

Contribution

It presents the first detailed spectral analysis of a unitary almost-Mathieu operator, establishing duality, computing Lyapunov exponents, and characterizing spectral types for various parameters.

Findings

Spectral type is purely absolutely continuous in subcritical region.

Spectral type is pure point in supercritical region.

Spectrum is a zero-measure Cantor set in the critical case.

Abstract

We introduce a unitary almost-Mathieu operator, which is obtained from a two-dimensional quantum walk in a uniform magnetic field. We exhibit a version of Aubry--Andr\'{e} duality for this model, which partitions the parameter space into three regions: a supercritical region and a subcritical region that are dual to one another, and a critical regime that is self-dual. In each parameter region, we characterize the cocycle dynamics of the transfer matrix cocycle generated by the associated generalized eigenvalue equation. In particular, we show that supercritical, critical, and subcritical behavior all occur in this model. Using Avila's global theory of one-frequency cocycles, we exactly compute the Lyapunov exponent on the spectrum in terms of the given parameters. We also characterize the spectral type for each value of the coupling constant, almost every frequency, and almost every phase. Namely, we show that for almost every frequency and every phase the spectral type is purely absolutely continuous in the subcritical region, pure point in the supercritical region, and purely singular continuous in the critical region. In some parameter regions, we refine the almost-sure results. In the critical case for instance, we show that the spectrum is a Cantor set of zero Lebesgue measure for arbitrary irrational frequency and that the spectrum is purely singular continuous for all but countably many phases.