[Paper Review] Anderson Localization for Quasi-Periodic CMV Matrices and Quantum Walks

Fengpeng Wang, David Damanik|arXiv (Cornell University)|Apr 1, 2018

Spectral Theory in Mathematical Physics8 references3 citations

TL;DR

This paper establishes Anderson localization for one-dimensional CMV matrices and quantum walks with analytic quasi-periodic Verblunsky coefficients or coins, proving pure point spectrum with exponentially decaying eigenfunctions in the regime of positive Lyapunov exponents. The result extends Bourgain and Goldstein's Schrödinger operator result to the unitary CMV setting using dynamical systems and Lyapunov exponent analysis.

ABSTRACT

We consider CMV matrices, both standard and extended, with analytic quasi-periodic Verblunsky coefficients and prove Anderson localization in the regime of positive Lyapunov exponents. This establishes the CMV analog of a result Bourgain and Goldstein proved for discrete one-dimensional Schrödinger operators. We also prove a similar result for quantum walks on the integer lattice with suitable analytic quasi-periodic coins.

Motivation & Objective

• To establish Anderson localization for CMV matrices with analytic quasi-periodic Verblunsky coefficients, extending results from Schrödinger operators to the unitary CMV setting.
• To analyze the spectral properties of extended CMV matrices and quantum walks under analytic quasi-periodic dynamics.
• To prove that positive Lyapunov exponents imply pure point spectrum with exponentially decaying eigenfunctions in the CMV context.
• To address a longstanding open problem in orthogonal polynomials on the unit circle by treating quasi-periodic Verblunsky coefficients systematically.

Proposed method

• Uses the dynamical systems approach to analyze the cocycle over the shift on the torus generated by the quasi-periodic coefficients.
• Applies the theory of Lyapunov exponents and acceleration to characterize spectral types in the positive exponent regime.
• Employs the resolvent identity and paving property to control Green's function decay and establish localization.
• Leverages the connection between CMV matrices and orthogonal polynomials on the unit circle (OPUC) via Verblunsky coefficients.
• Utilizes the unitary structure of CMV matrices and quantum walk evolution operators to derive spectral estimates.
• Applies compactness arguments and uniform bounds on the Lyapunov exponent over analytic parameter sets.

Experimental results

Research questions

• RQ1 Does Anderson localization occur for CMV matrices with analytic quasi-periodic Verblunsky coefficients when the Lyapunov exponent is positive?
• RQ2 Can the spectral localization result for Schrödinger operators be extended to the unitary CMV matrix setting?
• RQ3 What is the spectral type of quantum walks on the integer lattice with analytic quasi-periodic coins?
• RQ4 How do Lyapunov exponents and dynamical systems methods characterize localization in the CMV framework?

Key findings

• For CMV matrices with analytic quasi-periodic Verblunsky coefficients, the spectrum is purely pure point with exponentially decaying eigenfunctions when the Lyapunov exponent is positive.
• The Lyapunov exponent satisfies a uniform lower bound of the form $ L(\omega, S^{(E,\lambda v)}) \geq -\frac{1}{2}\ln(1-\lambda) + c_0 $ for $ \lambda \in (0,1) $, ensuring positivity in the relevant regime.
• The paving property is used to show that if Green's functions decay exponentially on small intervals, they also decay exponentially on larger intervals, implying localization.
• For quantum walks with analytic quasi-periodic coins, a similar Anderson localization result holds, establishing pure point spectrum with localized eigenstates.
• The proof relies on the acceleration theory of cocycles and uniform control over analytic parameter sets, leveraging compactness and analyticity.
• The result confirms the expected spectral behavior in the quasi-periodic CMV setting, analogous to the Schrödinger case, as conjectured by Simon.