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An exactly solvable spin chain related to Hahn polynomials

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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 α and depends on the parity of the chain site. Extending the model by a second parameter β, 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 (α,β) and (α+1,β−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.
Keywords
Quantum communication, perfect state transfer, Hahn polynomials

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Citation

Chicago
Stoilova, Nedialka, and Joris Van der Jeugt. 2011. “An Exactly Solvable Spin Chain Related to Hahn Polynomials.” Symmetry Integrability and Geometry-methods and Applications 7.
APA
Stoilova, Nedialka, & Van der Jeugt, J. (2011). An exactly solvable spin chain related to Hahn polynomials. SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS, 7.
Vancouver
1.
Stoilova N, Van der Jeugt J. An exactly solvable spin chain related to Hahn polynomials. SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS. 2011;7.
MLA
Stoilova, Nedialka, and Joris Van der Jeugt. “An Exactly Solvable Spin Chain Related to Hahn Polynomials.” SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS 7 (2011): n. pag. Print.
@article{1204699,
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 \ensuremath{\alpha} and depends on the parity of the chain site. Extending the model by a second parameter \ensuremath{\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 (\ensuremath{\alpha},\ensuremath{\beta}) and (\ensuremath{\alpha}+1,\ensuremath{\beta}\ensuremath{-}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.},
articleno    = {033},
author       = {Stoilova, Nedialka and Van der Jeugt, Joris},
issn         = {1815-0659},
journal      = {SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS},
keyword      = {Quantum communication,perfect state transfer,Hahn polynomials},
language     = {eng},
pages        = {13},
title        = {An exactly solvable spin chain related to Hahn polynomials},
url          = {http://dx.doi.org/10.3842/SIGMA.2011.033},
volume       = {7},
year         = {2011},
}


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