An Exactly Solvable Spin Chain Related to Hahn Polynomials

Neli I. Stoilova ; Joris Van der Jeugt

Symmetry, integrability and geometry: methods and applications, Tome 7 (2011)

Résumé

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.

Détail

Keywords: linear spin chain; Hahn polynomial; state transfer.

Bibliographie

[1] Shi T., Li Y., Song A., Sun C. P., “Quantum-state transfer via the ferromagnetic chain in a spatially modulated field”, Phys. Rev. A, 71 (2005), 032309, 5 pp., arXiv: quant-ph/0408152 | DOI

[2] Bose S., “Quantum communication through an unmodulated spin chain”, Phys. Rev. Lett., 91 (2003), 207901, 4 pp., arXiv: quant-ph/0212041 | DOI

[3] Bose S., Jin B.-Q., Korepin V. E., “Quantum communication through a spin ring with twisted boundary conditions”, Phys. Rev. A, 72 (2005), 022345, 4 pp., arXiv: quant-ph/0409134 | DOI

[4] Bose S., “Quantum communication through spin chain dynamics: an introductory overview”, Contemp. Phys., 48 (2007), 13–30, arXiv: 0802.1224 | DOI

[5] Lieb E., Wu F., “Absence of Mott transition in an exact solution of short-range 1-band model in 1 dimension”, Phys. Rev. Lett., 20 (1968), 1445–1448 | DOI

[6] Jordan P., Wigner E., “About the Pauli exclusion principle”, Z. Phys., 47 (1928), 631–651 | DOI | Zbl

[7] Christandl M., Datta N., Ekert A., Landahl A. J., “Perfect state transfer in quantum spin networks”, Phys. Rev. Lett., 92 (2004), 187902, 4 pp., arXiv: quant-ph/0309131 | DOI | MR

[8] Albanese C., Christandl M., Datta N., Ekert A., “Mirror inversion of quantum states in linear registers”, Phys. Rev. Lett., 93 (2004), 230502, 4 pp., arXiv: quant-ph/0405029 | DOI | MR

[9] Christandl M., Datta N., Dorlas T. C., Ekert A., Kay A., Landahl A. J., “Perfect transfer of arbitrary states in quantum spin networks”, Phys. Rev. A, 71 (2005), 032312, 11 pp., arXiv: quant-ph/0411020 | DOI

[10] Yung M. H., Bose S., “Perfect state transfer, effective gates, and entanglement generation in engineered bosonic and fermionic networks”, Phys. Rev. A, 71 (2005), 032310, 6 pp., arXiv: quant-ph/0407212 | DOI

[11] Karbach P., Stolze J., “Spin chains as perfect quantum state mirrors”, Phys. Rev. A, 72 (2005), 030301, 4 pp., arXiv: quant-ph/0501007 | DOI | MR

[12] Kay A., “A review of perfect state transfer and its application as a constructive tool”, Int. J. Quantum Inf., 8 (2010), 641–676, arXiv: 0903.4274 | DOI | Zbl

[13] Chakrabarti R., Van der Jeugt J., “Quantum communication through a spin chain with interaction determined by a Jacobi matrix”, J. Phys. A: Math. Theor., 43 (2010), 085302, 20 pp., arXiv: 0912.0837 | DOI | MR | Zbl

[14] Jafarov E. I., Van der Jeugt J., “Quantum state transfer in spin chains with $q$-deformed interaction terms”, J. Phys. A: Math. Theor., 43 (2010), 405301, 18 pp., arXiv: 1005.2912 | DOI | MR | Zbl

[15] Koekoek R., Lesky P. A., Swarttouw R. F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010 | DOI | MR | Zbl

[16] Nikiforov A. F., Suslov S. K., Uvarov V. B., Classical orthogonal polynomials of a discrete variable, Springer Series in Computational Physics, Springer-Verlag, Berlin, 1991 | MR

[17] Regniers G., Van der Jeugt J., “Analytically solvable Hamiltonians for quantum systems with a nearest-neighbour interaction”, J. Phys. A: Math. Theor., 42 (2009), 125301, 16 pp., arXiv: 0902.2308 | DOI | MR | Zbl

[18] Bailey W. N., Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, 32, Stechert-Hafner, Inc., New York, 1964 | MR

[19] Slater L. J., Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966 | MR | Zbl

[20] Qian X.-F., Li Y., Li Y., Song Z., Sun C. P., “Quantum-state transfer characterized by mode entanglement”, Phys. Rev. A, 72 (2005), 062329, 6 pp. | DOI

[21] Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96, 2nd ed., Cambridge University Press, Cambridge, 2004 | MR | Zbl

[22] Atakishiyev N. M., Pogosyan G. S., Wolf K. B., “Finite models of the oscillator”, Phys. Part. Nuclei, 36 (2005), 247–265