An Exactly Solvable Spin Chain Related to Hahn Polynomials

TL;DR

This paper introduces an exactly solvable spin chain model with parameters, where eigenstates are expressed via Hahn polynomials, enabling explicit calculations of correlation functions and exploring q-extensions.

Contribution

The paper extends a known spin chain model by adding a second parameter, deriving explicit eigenstates using Hahn polynomials, and establishing new difference equations for these polynomials.

Findings

Eigenstates expressed in terms of Hahn polynomials with parameters (lpha,eta) and (lpha+1,eta-1)

Closed-form correlation functions derived from explicit eigenstates

Discussion of a q-extension of the model

Abstract

We study a linear spin chain which was originally introduced by Shi et al. [Phys. Rev. A 71 (2005), 032309, 5 pages], for which the coupling strength contains a parameter $\alpha$ and depends on the parity of the chain site. Extending the model by a second parameter $\beta$, it is shown that the single fermion eigenstates of the Hamiltonian can be computed in explicit form. The components of these eigenvectors turn out to be Hahn polynomials with parameters $(\alpha,\beta)$ and $(\alpha+1,\beta-1)$. The construction of the eigenvectors relies on two new difference equations for Hahn polynomials. The explicit knowledge of the eigenstates leads to a closed form expression for the correlation function of the spin chain. We also discuss some aspects of a $q$-extension of this model.