An Exactly Solvable Spin Chain Related to Hahn Polynomials

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.

References in corpus (3)

• Perfect Transfer of Arbitrary States in Quantum Spin Networks
• Mirror Inversion of Quantum States in Linear Registers
• Spin Chains as Perfect Quantum State Mirrors

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