Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 019, 10 pages      nlin.SI/0602010      http://dx.doi.org/10.3842/SIGMA.2006.019

Eigenvectors of Open Bazhanov-Stroganov Quantum Chain

Nikolai Iorgov
Bogolyubov Institute for Theoretical Physics, 14b Metrolohichna Str., Kyiv, 03143 Ukraine

Received November 29, 2005, in final form January 30, 2006; Published online February 04, 2006

Abstract
In this contribution we give an explicit formula for the eigenvectors of Hamiltonians of open Bazhanov-Stroganov quantum chain. The Hamiltonians of this quantum chain is defined by the generation polynomial An(λ) which is upper-left matrix element of monodromy matrix built from the cyclic L-operators. The formulas for the eigenvectors are derived using iterative procedure by Kharchev and Lebedev and given in terms of wp(s)-function which is a root of unity analogue of Γq-function.

Key words: quantum integrable systems; Bazhanov-Stroganov quantum chain.

pdf (234 kb)   ps (174 kb)   tex (13 kb)

References

  1. Bazhanov V.V., Stroganov Yu.G., Chiral Potts model as a descendant of the six-vertex model, J. Statist. Phys., 1990, V.59, 799-817.
  2. Korepanov I.G., Hidden symmetries in the 6-vertex model, Chelyabinsk Polytechnical Inst., VINITI No. 1472-V87, 1987 (in Russian).
    Korepanov I.G., Hidden symmetries in the 6-vertex model of statistical physics, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 1994, V.215, 163-177 (English transl.: J. Math. Sci. (New York), 1997, V.85, 1661-1670), hep-th/9410066.
  3. Baxter R.J., Superintegrable chiral Potts model: thermodynamic properties, an "inverse" model, and a simple associated Hamiltonian, J. Statist. Phys., 1989, V.57, 1-39.
  4. Baxter R.J., Bazhanov V.V., Perk J.H.H., Functional relations for the transfer matrices of the chiral Potts model, Internat. J. Modern Phys. B, 1990, V.4, 803-869.
  5. Baxter R.J., Chiral Potts model: eigenvalues of the transfer matrix, Phys. Lett. A, 1990, V.146, 110-114.
  6. Baxter R.J., Derivation of the order parameter of the chiral Potts model, Phys. Rev. Lett., 2005, V.94, 130602, 3 pages, cond-mat/0501227.
  7. Kharchev S., Lebedev D., Eigenfunctions of GL(N,R) Toda chain: the Mellin-Barnes representation, JETP Lett., 2000, V.71, 235-238, hep-th/0004065.
  8. Sklyanin E.K., Quantum inverse scattering method. Selected topics, in Nankai Lectures in Mathematical Physics "Quantum Group and Quantum Integrable Systems", Editor Mo-Lin Ge, Singapore, World Scientific, 1992, 63-97.
  9. Kharchev S., Lebedev D., Semenov-Tian-Shansky M., Unitary representations of Uq(sl(2,R)), the modular double, and the multiparticle q-deformed Toda chains, Comm. Math. Phys., 2002, V.225, N 3, 573-609, hep-th/0102180.
  10. Iorgov N., Shadura V., Wave functions of the Toda chain with boundary interaction, Theoret. and Math. Phys., 2005, V.142, N 2, 289-305, nonlin.SI/0411002.
  11. von Gehlen G., Iorgov N., Pakuliak S., Shadura V., Baxter-Bazhanov-Stoganov model: separation of variables and Baxter equation, in preparation.
  12. Bugrij A.I., Iorgov N.Z., Shadura V.N., Alternative method of calculating the eigenvalues of the transfer matrix of the t2 model for N=2, JETP Lett., 2005, V.82, 311-315.
  13. Lisovyy O., Transfer matrix eigenvectors of the Baxter-Bazhanov-Stroganov t2-model for N=2, J. Phys. A: Math. Gen., 2006, V.39, to appear, nlin.SI/0512026.
  14. Tarasov V.O., Cyclic monodromy matrices for the R-matrix of the six-vertex model and the chiral Potts model with fixed spin boundary conditions, Internat. J. Modern Phys. A Suppl., 1992, V.7, 963-975.
  15. Pakuliak S., Sergeev S., Quantum relativistic Toda chain at root of unity: isospectrality, modified Q-operator and functional Bethe ansatz, Int. J. Math. Math. Sci., 2002, V.31, 513-554, nlin.SI/0205037.
  16. von Gehlen G., Pakuliak S., Sergeev S., The Bazhanov-Stroganov model from 3D approach, J. Phys. A: Math. Gen., 2005, V.38, 7269-7298, nlin.SI/0505019.
  17. Bazhanov V.V., Baxter R.J., Star-triangle relation for a three dimensional model, J. Statist. Phys., 1993, V.71, 839-864, hep-th/9212050.

Previous article   Next article   Contents of Volume 2 (2006)