Spectral Characteristics of the Unitary Critical Almost-Mathieu Operator

TL;DR

This paper analyzes the spectral properties of a quantum walk model with quasi-periodic coins, revealing it has a Cantor spectrum of zero measure, no absolutely continuous or point spectrum, and exhibits critical spectral behavior similar to the Almost-Mathieu Operator.

Contribution

It demonstrates that the unitary operator shares spectral features with the critical Almost-Mathieu Operator, including singular cocycles and Cantor spectrum, extending Avila's global theory to this setting.

Findings

Empty absolutely continuous spectrum

Zero Lebesgue measure Cantor spectrum

Empty point spectrum for irrational frequencies

Abstract

We discuss spectral characteristics of a one-dimensional quantum walk whose coins are distributed quasi-periodically. The unitary update rule of this quantum walk shares many spectral characteristics with the critical Almost-Mathieu Operator; however, it possesses a feature not present in the Almost-Mathieu Operator, namely singularity of the associated cocycles (this feature is, however, present in the so-called Extended Harper's Model). We show that this operator has empty absolutely continuous spectrum and that the Lyapunov exponent vanishes on the spectrum; hence, this model exhibits Cantor spectrum of zero Lebesgue measure for all irrational frequencies and arbitrary phase, which in physics is known as Hofstadter's butterfly. In fact, we will show something stronger, namely, that all spectral parameters in the spectrum are of critical type, in the language of Avila's global theory of analytic quasiperiodic cocycles. We further prove that it has empty point spectrum for each irrational frequency and away from a frequency-dependent set of phases having Lebesgue measure zero. The key ingredients in our proofs are an adaptation of Avila's Global Theory to the present setting, self-duality via the Fourier transform, and a Johnson-type theorem for singular dynamically defined CMV matrices which characterizes their spectra as the set of spectral parameters at which the associated cocycles fail to admit a dominated splitting.