Spectral Transitions and Singular Continuous Spectrum in A New Family of Quasi-periodic Quantum Walks

TL;DR

This paper introduces a new class of quasi-periodic quantum walks modeled by extended CMV matrices, revealing a stable region with purely singular continuous spectrum, thus advancing understanding of spectral properties in quantum dynamics.

Contribution

It presents the first solvable quasi-periodic quantum walk model with a stable purely singular continuous spectrum, generalizing the unitary almost Mathieu operator.

Findings

Reveals a richer spectral phase diagram similar to the extended Harper's model

Provides the first example of a solvable quasi-periodic quantum walk with singular continuous spectrum

Establishes a connection between quantum walks and extended CMV matrices

Abstract

This paper introduces and rigorously analyzes a new class of one-dimensional discrete-time quantum walks whose dynamics are governed by a parametrized family of extended CMV matrices. The model generalizes the unitary almost Mathieu operator (UAMO) and exhibits a richer spectral phase diagram, closely resembling the extended Harper's model. It provides the first example of a solvable quasi-periodic quantum walk that exhibits a stable region of purely singular continuous spectrum.