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Bethe's Equation Is Incomplete for the XXZ Model at Roots of Unity

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Abstract

We demonstrate for the six vertex and XXZ model parameterized by Δ= −(q+q-1)/2≠±1 that when q2N=1 for integer N≥2 the Bethe's ansatz equations determine only the eigenvectors which are the highest weights of the infinite dimensional sl2 loop algebra symmetry group of the model. Therefore in this case the Bethe's ansatz equations are incomplete and further conditions need to be imposed in order to completely specify the wave function. We discuss how the evaluation parameters of the finite dimensional representations of the sl2 loop algebra can be used to complete this specification.

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REFERENCES

  1. H. A. Bethe, Zur Theorie der Metalle, Z. Physik 71:205 (1931).

    Google Scholar 

  2. R. Orbach, Linear antiferromagnetic chain with anisotropic coupling, Phys. Rev. 112:309 (1958).

    Google Scholar 

  3. L. R. Walker, Antiferromagnetic linear chain, Phys. Rev. 116:1089 (1959).

    Google Scholar 

  4. J. des Cloizeaux and M Gaudin, Anisotropic linear magnetic chain, J. Math. Phys. 7:1384 (1966).

    Google Scholar 

  5. C. N. Yang and C. P. Yang, One-dimensional chain of anisotropic spin-spin interactions I. Proof of Bethe's hypothesis for ground state in a finite system, Phys. Rev. 150:321 (1966).

    Google Scholar 

  6. C. N. Yang and C. P. Yang, One-dimensional chain of anisotropic spin-spin interactions. II. Properties of the ground-state energy per lattice site for an infinite system, Phys. Rev. 150:327 (1966).

    Google Scholar 

  7. M. Takahashi, One dimensional Heisenberg model at finite temperature, Prog. Theo. Phys. 46:401 (1971).

    Google Scholar 

  8. L. D. Faddeev and L. Takhtajan, What is the spin of a spin wave? Phys. Lett. 85A:375 (1981).

    Google Scholar 

  9. L. D. Faddeev and L. Takhtajan, The spectrum and scattering of excitations in the one dimensional isotropic Heisenberg model, J. Sov. Math. 24:241 (1984).

    Google Scholar 

  10. A. N. Kirillov, Combinatorial identities and the completeness of states for Heisenberg magnet, J. Sov. Math. 30:2298 (1985).

    Google Scholar 

  11. A. N. Kirillov, Completeness of the states of the generalized Heisenberg model, J. Sov. Math. 36:115 (1987).

    Google Scholar 

  12. A. Klümper and J. Zittartz, Eigenvalues of the eight-vertex model and the spectrum of the XYZ Hamiltonian, Z. Phys. B 71:495 (1988).

    Google Scholar 

  13. A. Klümper and J. Zittartz, The eight-vertex model: spectrum of the transfer matrix and classification of the excited states, Z. Phys. B 75:371 (1989).

    Google Scholar 

  14. R. P. Langlands and Y. Saint-Aubin, Algebro-geometric aspects of the Bethe equations, in Proceedings of the Gursey Memorial Conference I-Strings and Symmetries, G. Atkas, C. Saclioglu, and M. Serdaroglu, eds., Lecture Notes in Physics, Vol. 447 (Springer Verlag, 1994).

  15. V. Tarasov and A. Varchenko, Bases of Bethe vectors and difference equations with regular singular points, International Mathematics Research Notes 13:637 (1995), q-alg/ 9504011.

    Google Scholar 

  16. R. P. Langlands and Y. Saint-Aubin, Aspects combinatoires des equations de Bethe, in Advances in Mathematical Sciences: CRM's 25 five years, Luc Vinet, ed., CRM Proceedings and Lecture Notes, Vol. 11 (Am. Math. Soc., 1997), p. 231.

  17. A. N. Kirillov and N. A. Liskova, Completeness of Bethe's states for the generalized XXZ model, J. Phys. A 30:1209 (1997).

    Google Scholar 

  18. R. Siddharthan, Singularities in the Bethe solution of the XXX and the XXZ Heisenberg spin chains, cond-mat/9804210.

  19. A. Ilakovac, M. Kolanivic, S. Pullua, and P. Prester, Violation of the string hypothesis and the Heisenberg spin chain, Phys. Rev. B 60:7271 (1999).

    Google Scholar 

  20. J. D. Noh, D. S. Lee, and D. Kim, Incompleteness of regular solutions of the Bethe ansatz for Heisenberg XXZ spin chain, condmat/0001175.

  21. M. Takahashi and M. Suzuki, One-dimensional anisotropic Heisenberg model at finite temperatures, Prog. Theo. Phys. 48:2187 (1972).

    Google Scholar 

  22. F. H. L. Essler, V. E. Korepin, and K. Schoutens, Fine structure of the Bethe ansatz for the spin-\({\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}\) Heisenberg XXX model, J. Phys. A 25:4115 (1992).

    Google Scholar 

  23. M. Takahashi, Thermodynamic Bethe ansatz and condensed matter, cond-mat/9708087.

  24. F. Woynarovich, On the S z=0 excited states of an anisotropic Heisenberg chain, J. Phys. A 15:2985 (1982).

    Google Scholar 

  25. C. Destri amd J. H. Lowenstein, Analysis of the Bethe-Ansatz equations of the chiralinvariant Gross–Neveu model, Nucl. Phys. B 205[FS5]:369 (1982).

    Google Scholar 

  26. O. Babelon, H. J. de Vega, and C. M. Viallet, Analysis of the Bethe ansatz equations of the XXZ model, Nucl. Phys. B 220[FS8]:13 (1983).

    Google Scholar 

  27. A. A. Vladimirov, Non-string two-magnon configurations in the isotropic Heisenberg magnet, Phys. Lett. 105A:418 (1984).

    Google Scholar 

  28. R. J. Baxter, Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain I: Some fundamental eigenvectors, Ann. Phys. 76:1 (1973).

    Google Scholar 

  29. R. J. Baxter, Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain II: Equivalence to a generalized ice-type lattice model, Ann. Phys. 76:25 (1973).

    Google Scholar 

  30. R. J. Baxter, Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain III: Eigenvectors of the transfer matrix and the Hamiltonian, Ann. Phys. 76:48 (1973).

    Google Scholar 

  31. I. G. Korepanov, Hidden symmetry of the six vertex model (Chelyab. Politekhn. Inst., Chelyabinsk, 1987), Manuscript No. 1472-887, deposited at VITITI (Moscow, Russian).

    Google Scholar 

  32. V. Pasquier and H. Saleur, Symmetries of the XXZ chain and quantum SU(2) in Fields, String and critical phenomena, Proc. Les Houches Summer School 1988 (North Holland).

  33. F. C. Alcaraz, U. Grimm, and V. Rittenberg, The XXZ Heisenberg chain, conformal invariance and the operator content of c < 1 systems, Nucl. Phys. B 316:735 (1989).

    Google Scholar 

  34. V. Pasquier and H. Saleur, Common structures between finite systems and conformal field theory through quantum groups, Nucl. Phys. B 330:523 (1990).

    Google Scholar 

  35. I. G. Korepanov, Vacuum curves of the L-operators related to the six-vertex model, St. Petersburg Math. J., 349 (1995).

  36. I. G. Korepanov, Hidden symmetries in the 6-vertex model of statistical physics, hepth/9410066.

  37. T. Deguchi, K. Fabricius, and B. M. McCoy, The sl 2 loop algebra symmetry of the six vertex model at roots of unity, J. Stat. Phys. (in press).

  38. E. H. Lieb, Exact solution of the problem of the entropy of two-dimensional ice, Phys. Rev. Letts. 18:692 (1967).

    Google Scholar 

  39. E. H. Lieb, Exact solution of the Fmodel of an antiferroelectric, Phys. Rev. Lett. 18:1046 (1967).

    Google Scholar 

  40. E. H. Lieb, Exact solution of the two-dimensional Slater KDP model of a ferroelectric, Phys. Rev. Lett. 19:108 (1967).

    Google Scholar 

  41. E. H. Lieb, Residual entropy of square ice, Phys. Rev. 162:162 (1967).

    Google Scholar 

  42. B. Sutherland, Exact solution of a two-dimensional model for hydrogen bonded crystals, Phys. Rev. Lett. 19:103 (1967).

    Google Scholar 

  43. C. P. Yang, Exact solution of two dimensional ferroelectrics in an arbitrary external field, Phys. Rev. Lett. 19:586 (1967).

    Google Scholar 

  44. B. Sutherland, C. N. Yang, and C. P. Yang, Exact solution of two dimensional ferroelectrics in an arbitrary external field, Phys. Rev. Lett. 19:588 (1967).

    Google Scholar 

  45. R. J. Baxter, Eight-vertex model in lattice statistics, Phys. Rev. Lett. 26:832 (1971); Onedimensional anisotropic Heisenberg chain, Phys. Rev. Lett. 26:834 (1971).

    Google Scholar 

  46. R. J. Baxter, Partition function of the eight vertex lattice model, Ann. Phys. 70:193 (1972).

    Google Scholar 

  47. E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Ann. Phys. 16:407 (1961).

    Google Scholar 

  48. V. Chari, Integrable representations of affine Lie-algebras, Inv. Math. 85:317 (1986).

    Google Scholar 

  49. V. Chari and A. Pressley, Quantum affine algebras, Comm. Math. Phys. 142:142 (1991).

    Google Scholar 

  50. V. Chari and A. Pressley, Quantum affine algebras at roots of unity, q-alg/9690031.

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Authors and Affiliations

  1. Physics Department, University of Wuppertal, 42097, Wuppertal, Germany

    Klaus Fabricius

  2. Institute for Theoretical Physics, State University of New York, Stony Brook, New York, 11794-3840

    Barry M. McCoy

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  1. Klaus Fabricius
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  2. Barry M. McCoy
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Fabricius, K., McCoy, B.M. Bethe's Equation Is Incomplete for the XXZ Model at Roots of Unity. Journal of Statistical Physics 103, 647–678 (2001). https://doi.org/10.1023/A:1010380116927

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