Arithmetic spectral transition for the unitary almost Mathieu operator

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 $0\le \hat{I}{»}_1<\hat{I}{»}_2\le 1$, we establish an arithmetic localization statement governed by the frequency exponent ${\hat{I}}^2(\ddot{I}‰)$. More precisely, for every irrational $\ddot{I}‰$ with ${\hat{I}}^2(\ddot{I}‰)<L$, where $L>0$ denotes the Lyapunov exponent, and every non-resonant phase $\hat{I}¸$, 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 ${\hat{I}}^2(\ddot{I}‰)=0$) to a sharp threshold in frequency.