Skip to main content
SpringerLink
Log in

Analytic Bethe ansatz for fundamental representations of Yangians

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study the analytic Bethe, ansatz in solvable vertex models associated with the YangianY(X r ) or its quantum affine analogueU q (X (1) r ) forX r =B r ,C r andD r . Eigenvalue formulas are proposed for the transfer matrices related to all the fundamental representations ofY(X r ). Under the Bethe ansatz equation, we explicitly prove that they are pole-free, a crucial property in the ansatz. Conjectures are also given on higher representation cases by applying theT-system, the transfer matrix functional relations proposed recently. The eigenvalues are neatly described in terms of Yangian analogues of the semi-standard Young tableaux.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bethe, H.A.: Zur Theorie der Metalle, I. Eigenwerte und Eigenfunktionen der linearen Atomkette. Z. Physik71, 205–226 (1931)

    Article  Google Scholar 

  2. Yang, C.N., Yang, C.P.: Thermodynamics of a one-dimensional, system of bosons with repulsive delta-function interaction. J. Math. Phys.10, 1115–1122 (1969)

    Article  Google Scholar 

  3. Sklyanin, E.K., Takhtajan, L.A., Faddeev, L.D.: Quantum inverse problem method I. Theor. Math. Phys.40, 688–706 (1980); Kulish, P.P., Sklyanin, E.K.: Quantum spectral transform method. Recent developments. Lect. Note. Phys.151, 61–119 Springer, 1982

    Article  Google Scholar 

  4. Baxter, R.J.: Partition function of the eight-vertex lattice model. Ann. Phys.70, 193–228 (1972)

    Article  Google Scholar 

  5. Reshetikhin, N.Yu.: The functional equation method in the theory of exactly soluble quantum systems. Sov. Phys. JETP57, 691–696 (1983)

    Google Scholar 

  6. Sklyanin, E.K.: Quantum inverse scattering method. Selected topics. In: Quantum Group and Quantum Integrable Systems. ed. Mo-Lin Ge, Singapore: World Scientific, 1992

    Google Scholar 

  7. Paz, R.: Feynman's office: The last blackboards. Phys. Today42, 88 (1989)

    Google Scholar 

  8. Kuniba, A., Nakanishi, T., Suzuki, J.: Functional relations in solvable lattice models I: Functional relations and representation theory. Int. J. Mod. Phys.A9 (1994) 5215–5266

    Article  Google Scholar 

  9. Kuniba, A., Nakanishi, T., Suzuki, J.: Functional relations in solvable lattice models II: Applications. Int. J. Mod. Phys.A9 (1994) 5267–5312

    Article  Google Scholar 

  10. Kuniba, A.: Analytic Bethe ansatz andT-system in C (1)2 vertex models. J. Phys. A.: Math. Gen.27, L113-L118 (1994)

    Article  Google Scholar 

  11. Suzuki, J.: FusionU q (G (1)2 ) vertex models and analytic Bethe ansätze. Phys. Lett.A195 (1994) 190–197

    Article  Google Scholar 

  12. Drinfel's, V.G.: Hopf algebras and quantum Yang-Baxter equation. Sov. Math. Dokl.32, 254–258 (1985)

    Google Scholar 

  13. Drinfel'd, V.G.: Quantum groups. ICM Proceedings, Berkeley, pp. 798–820 (1986)

  14. Jimbo, M.: Aq-difference analogue ofU(g) and the Yang-Baxter equation. Lett. Math. Phys.10, 63–69 (1985)

    Article  Google Scholar 

  15. Kac, V.G.: Infinite Dimensional Lie Algebras. Cambridge: Cambridge University Press, 1990

    Google Scholar 

  16. Kirillov, A. N., Reshetikhin, N.Yu.: Representations of Yangians and multiplicity of occurrence of the irreducible components of the tensor product of representations of simple Lie algebras. J. Sov. Math.52, 3156–3164 (1990)

    Google Scholar 

  17. Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. London: Academic Press, 1982

    Google Scholar 

  18. Kirillov, A.N., Reshetikhin, N.Yu.: Exact solution of the integrableXXZ Heisenberg model with arbitrary spin: I. The ground state and the excitation spectrum. J. Phys. A.: Math. Gen.20, 1565–1585 (1987)

    Article  Google Scholar 

  19. Bazhanov, V.V., Reshetikhin, N.Yu.: Restricted solid on solid models connected with simply laced algebras and conformal field theory. J. Phys. A: Math. Gen.23, 1477–1492 (1990)

    Article  Google Scholar 

  20. Reshetikhin, N.Yu.: Integrable models of quantum one-dimensional magnets withO(n) andSp(2k) symmetry. Theor. Math. Phys.63, 555–569 (1985)

    Article  Google Scholar 

  21. Reshetikhin, N.Yu.: The spectrum of the transfer matrices connected with Kac-Moody algebras. Lett. Math. Phys.14, 235–246 (1987)

    Article  Google Scholar 

  22. Klümper, A., Pearce, P.A.: Conformal weights of RSOS lattice models and their fusion hierarchies. PhysicaA183, 304–350 (1992)

    Article  Google Scholar 

  23. Jimbo, M., Kuniba, A., Miwa, T., Okado, M.: TheA (1) n face models. Commun. Math. Phys.119, 543–565 (1988)

    Article  Google Scholar 

  24. Kashiwara, M., Nakashima, T.: Crystal graphs for representations of theq-analogue of classical Lie algebras. RIMS preprint767 (1991)

  25. Nakashima, T.: Crystal base and a generalization of the Littlewood-Richardson rule for the classical Lie algebras. Commun. Math. Phys.154, 215–243 (1993)

    Article  Google Scholar 

  26. Ogievetsky, E.I., Wiegmann, P.B.: FactorizedS-matrix and the Bethe ansatz for simple Lie groups. Phys. Lett.B168, 360–366 (1986)

    Article  Google Scholar 

  27. Drinfel'd, V.G.: A new realization of Yangians and quantum affine algebras. Sov. Math. Dokl.36, 212–216 (1988)

    Google Scholar 

  28. Chari, V., Pressley, A.: Fundamental representations of Yangians and singularities ofR-matrices. J. reine angew. Math.,417, 87–128 (1991)

    Google Scholar 

  29. Kuniba, A., Nakanishi, T.: Spectra in conformal field theories from the Rodgers dilogarithm. Mod. Phys. Lett.A7, 3487–3494 (1992)

    Article  Google Scholar 

  30. Kirillov, A.N.: Identities for the Rodgers dilogarithm function connected with simple Lie algebras. J. Sov. Math.47, 2450–2459 (1989)

    Article  Google Scholar 

  31. Kuniba, A., Nakanishi, T., Suzuki, J.: Characters in conformal field theories from thermodynamic Bethe ansatz. Mod. Phys. Lett.A8, 1649–1659 (1993)

    Article  Google Scholar 

  32. Reshetikhin, N.Yu.: Algebraic Bethe ansatz forSO(n) invariant transfer-matrices. Zapiski nauch. LOMI169, 122–140 (1984) (in Russian); Okado, M.: QuantumR matrices related to the spin representation ofB r andD r . Commun. Math. Phys.134, 467–486 (1990)

    Google Scholar 

  33. Kuniba, A.: Thermodynamics of theU q (X (1) r ) Bethe ansatz system withq a root of unity. Nucl. Phys.B389, 209–244 (1993)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Physics, University of Tokyo, Komaba, 3-8-1, Meguro-ku, 153, Tokyo, Japan

    Atsuo Kuniba & Junji Suzuki

Authors
  1. Atsuo Kuniba
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Junji Suzuki
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by M. Jimbo

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kuniba, A., Suzuki, J. Analytic Bethe ansatz for fundamental representations of Yangians. Commun.Math. Phys. 173, 225–264 (1995). https://doi.org/10.1007/BF02101234

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02101234

Keywords

Search

Navigation