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

SIGMA 2 (2006), 021, 10 pages      cond-mat/0602427

On the Degenerate Multiplicity of the sl2 Loop Algebra for the 6V Transfer Matrix at Roots of Unity

Tetsuo Deguchi
Department of Physics, Faculty of Science, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-Ku, Tokyo 112-8610, Japan

Received October 31, 2005, in final form February 06, 2006; Published online February 17, 2006

We review the main result of cond-mat/0503564. The Hamiltonian of the XXZ spin chain and the transfer matrix of the six-vertex model has the sl2 loop algebra symmetry if the q parameter is given by a root of unity, q02N = 1, for an integer N. We discuss the dimensions of the degenerate eigenspace generated by a regular Bethe state in some sectors, rigorously as follows: We show that every regular Bethe ansatz eigenvector in the sectors is a highest weight vector and derive the highest weight dk±, which leads to evaluation parameters aj. If the evaluation parameters are distinct, we obtain the dimensions of the highest weight representation generated by the regular Bethe state.

Key words: loop algebra; the six-vertex model; roots of unity representations of quantum groups; Drinfeld polynomial.

pdf (249 kb)   ps (177 kb)   tex (14 kb)


  1. Alcaraz F.C., Grimm U., Rittenberg V., The XXZ Heisenberg chain, conformal invariance and the operator content of c < 1 systems, Nucl. Phys. B, 1989, V.316, 735-768.
  2. Baxter R.J., Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. I. Some fundamental eigenvectors, Ann. Phys., 1973, V.76, 1-24; II. Equivalence to a generalized Ice-type lattice model, Ann. Phys., 1973, V.76, 25-47; III. Eigenvectors of the transfer matrix and Hamiltonian, Ann. Phys., 1973, V.76, 48-71.
  3. Baxter R.J., Completeness of the Bethe ansatz for the six and eight vertex models, J. Statist. Phys., 2002, V.108, 1-48, cond-mat/0111188.
  4. Baxter R.J., The six and eight-vertex models revisited, J. Statist. Phys., 2004, V.116, 43-66, cond-mat/0403138.
  5. Braak D., Andrei N., On the spectrum of the XXZ-chain at roots of unity, J. Statist. Phys., 2001, V.105, 677-709, cond-mat/0106593.
  6. Chari V., Pressley A., Quantum affine algebras, Comm. Math. Phys., 1991, V.142, 261-283.
  7. Chari V., Pressley A., Quantum affine algebras at roots of unity, Represent. Theory, 1997, V.1, 280-328, q-alg/9609031.
  8. Chari V., Pressley A., Weyl modules for classical and quantum affine algebras, Represent. Theory, 2001, V.5, 191-223, math.QA/0004174.
  9. Deguchi T., Construction of some missing eigenvectors of the XYZ spin chain at the discrete coupling constants and the exponentially large spectral degeneracy of the transfer matrix, J. Phys. A: Math. Gen., 2002, V.35, 879-895, cond-mat/0109078.
  10. Deguchi T., The 8V CSOS model and the sl2 loop algebra symmetry of the six-vertex model at roots of unity, Internat. J. Modern Phys. B, 2002, V.16, 1899-1905, cond-mat/0110121.
  11. Deguchi T., XXZ Bethe states as highest weight vectors of the sl2 loop algebra at roots of unity, cond-mat/0503564.
  12. Deguchi T., The six-vertex model at roots of unity and some highest weight representations of the sl2 loop algebra, in preparation (to be submitted to the Proceedings of RAQIS'05, Annecy, France).
  13. Deguchi T., Fabricius K., McCoy B.M., The sl2 loop algebra symmetry of the six-vertex model at roots of unity, J. Statist. Phys., 2001, V.102, 701-736, cond-mat/9912141.
  14. Fabricius K., McCoy B.M., Bethe's equation is incomplete for the XXZ model at roots of unity, J. Statist. Phys., 2001, V.103, 647-678, cond-mat/0009279.
  15. Fabricius K., McCoy B.M., Completing Bethe's equations at roots of unity, J. Statist. Phys., 2001, V.104, 573-587, cond-mat/0012501.
  16. Fabricius K., McCoy B.M., Evaluation parameters and Bethe roots for the six-vertex model at roots of unity, Progress in Mathematical Physics, Vol. 23 (MathPhys Odyssey 2001), Editors M. Kashiwara and T. Miwa, Boston, Birkhäuser, 2002, 119-144, cond-mat/0108057.
  17. Fabricius K., McCoy B.M., New developments in the eight-vertex model, J. Statist Phys., 2003, V.111, 323-337, cond-mat/0207177.
    Fabricius K., McCoy B.M., Functional equations and fusion matrices for the eight-vertex model, Publ. Res. Inst. Math. Sci., 2004, V.40, 905-932, cond-mat/0311122.
  18. Fabricius K., McCoy B.M., New developments in the eight-vertex model II. Chains of odd length, cond-mat/0410113.
  19. Jimbo M., Private communication, July 2004.
  20. Kac V., Infinite dimensional Lie algebras, Cambridge, Cambridge University Press, 1990.
  21. 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.
  22. Korepanov I.G., Vacuum curves of the L-operators related to the six-vertex model, St. Petersburg Math. J., 1995, V.6, 349-364.
  23. Korepin V.E., Bogoliubov N.M., Izergin A.G., Quantum inverse scattering method and correlation functions, Cambridge, Cambridge University Press, 1993.
  24. Korff C., McCoy B.M., Loop symmetry of integrable vertex models at roots of unity, Nucl. Phys. B, 2001, V.618, 551-569, hep-th/0104120.
  25. Lusztig G., Modular representations and quantum groups, Contemp. Math., 1989, V.82, 59-77.
  26. Lusztig G., Introduction to quantum groups, Boston, Birkhäuser, 1993.
  27. Pasquier V., Saleur H., Common structures between finite systems and conformal field theories through quantum groups, Nucl. Phys. B, 1990, V.330, 523-556.
  28. Takhtajan L., Faddeev L., Spectrum and scattering of excitations in the one-dimensional isotropic Heisenberg model, J. Sov. Math., 1984, V.24, 241-267.
  29. 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, in Infinite Analysis, Part A, B (Kyoto, 1991), Adv. Ser. Math. Phys., Vol. 16, River Edge, NJ, World Sci. Publishing, 1992, 963-975.
  30. Tarasov V.O., On the Bethe vectors for the XXZ model at roots of unity, math.QA/0306032.

Previous article   Next article   Contents of Volume 2 (2006)