Quantum communication through a spin chain with interaction determined by a Jacobi matrix

TL;DR

This paper analyzes the time evolution of a single spin excitation in a linear spin chain with Hamiltonian related to Jacobi matrices of orthogonal polynomials, providing explicit correlation functions for various polynomial cases.

Contribution

It derives explicit correlation functions for spin chains with Hamiltonians linked to Jacobi matrices of orthogonal polynomials, including new closed-form solutions and asymptotic behaviors.

Findings

Closed-form correlation functions for Krawtchouk polynomials

Asymptotic correlation functions related to Charlier polynomials

Correlation functions for Hahn, dual Hahn, and Racah polynomials

Abstract

We obtain the time-dependent correlation function describing the evolution of a single spin excitation state in a linear spin chain with isotropic nearest-neighbour XY coupling, where the Hamiltonian is related to the Jacobi matrix of a set of orthogonal polynomials. For the Krawtchouk polynomial case an arbitrary element of the correlation function is expressed in a simple closed form. Its asymptotic limit corresponds to the Jacobi matrix of the Charlier polynomial, and may be understood as a unitary evolution resulting from a Heisenberg group element. Correlation functions for Hamiltonians corresponding to Jacobi matrices for the Hahn, dual Hahn and Racah polynomials are also studied. For the Hahn polynomials we obtain the general correlation function, some of its special cases, and the limit related to the Meixner polynomials, where the su(1,1) algebra describes the underlying symmetry. For the cases of dual Hahn and Racah polynomials the general expressions of the correlation functions contain summations which are not of hypergeometric type. Simplifications, however, occur in special cases.