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

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.

23 pages

References in corpus (3)

• Asymptotic evolution of quantum walks with random coin
• Properties of 1D classical and quantum Ising quasicrystals: rigorous results
• Localization of discrete-time quantum walks on a half line via the CGMV method

Cited by in corpus (5)

• The Fibonacci Hamiltonian
• Spectral Characteristics of the Unitary Critical Almost-Mathieu Operator
• Purely singular continuous spectrum for CMV operators generated by subshifts
• Resolvent Methods for Quantum Walks with an Application to a Thue-Morse Quantum Walk
• Spectral Properties of Continuum Fibonacci Schrödinger Operators