Orthogonal Polynomials on the Unit Circle with Fibonacci Verblunsky Coefficients, II. Applications

TL;DR

This paper explores spectral properties of CMV matrices with Fibonacci-based Verblunsky coefficients, applying the theory to quantum walks and the Ising model, revealing explicit spreading rate estimates and spectral-zero correspondences.

Contribution

It introduces new applications of Fibonacci Verblunsky coefficients in spectral analysis, linking quantum dynamics and statistical mechanics models.

Findings

Explicit estimates for quantum walk spreading rates.

Connection between Lee-Yang zeros and CMV spectrum.

Support of zeros matches the spectrum of constructed CMV matrices.

Abstract

We consider CMV matrices with Verblunsky coefficients determined in an appropriate way by the Fibonacci sequence and present two applications of the spectral theory of such matrices to problems in mathematical physics. In our first application we estimate the spreading rates of quantum walks on the line with time-independent coins following the Fibonacci sequence. The estimates we obtain are explicit in terms of the parameters of the system. In our second application, we establish a connection between the classical nearest neighbor Ising model on the one-dimensional lattice in the complex magnetic field regime, and CMV operators. In particular, given a sequence of nearest-neighbor interaction couplings, we construct a sequence of Verblunsky coefficients, such that the support of the Lee-Yang zeros of the partition function for the Ising model in the thermodynamic limit coincides with the essential spectrum of the CMV matrix with the constructed Verblunsky coefficients. Under certain technical conditions, we also show that the zeros distribution measure coincides with the density of states measure for the CMV matrix.