Arithmetic spectral transition for the unitary almost Mathieu operator

TL;DR

This paper investigates the spectral properties of the unitary almost Mathieu operator, demonstrating Anderson localization under certain arithmetic conditions on the frequency, extending previous results to a broader class of frequencies.

Contribution

It establishes a sharp threshold for localization in the UAMO based on the frequency's arithmetic properties, generalizing prior Diophantine frequency results.

Findings

Proves Anderson localization for UAMO with non-resonant phases and frequencies below a certain arithmetic threshold.

Extends localization results from Diophantine to more general irrational frequencies.

Identifies a sharp arithmetic criterion involving the frequency exponent for localization.

Abstract

We study the unitary almost Mathieu operator (UAMO), a one-dimensional quasi-periodic unitary operator arising from a two-dimensional discrete-time quantum walk on $\mathbb Z^2$ in a homogeneous magnetic field. In the positive Lyapunov exponent regime $0\le \lambda_1<\lambda_2\le 1$, we establish an arithmetic localization statement governed by the frequency exponent $\beta(\omega)$. More precisely, for every irrational $\omega$ with $\beta(\omega)<L$, where $L>0$ denotes the Lyapunov exponent, and every non-resonant phase $\theta$, we prove Anderson localization, i.e. pure point spectrum with exponentially decaying eigenfunctions. This extends our previous arithmetic localization result for Diophantine frequencies (for which $\beta(\omega)=0$) to a sharp threshold in frequency.