An Exactly Solvable Spin Chain Related to Hahn Polynomials

Details An Exactly Solvable Spin Chain Related to Hahn Polynomials An Exactly Solvable Spin Chain Related to Hahn Polynomials

2011-01-24

arxiv.org/abs/1101.4469v2

SIGMA 7 (2011), 033, 13 pages

Physics

Mathematical Physics

Scientific paper

10.3842/SIGMA.2011.033

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.

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