Skip to main content
Log in

Q-Operators for Higher Spin Eight Vertex Models with an Even Number of Sites

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

Abstract

We construct the Q-operator for generalised eight vertex models associated to higher spin representations of the Sklyanin algebra, following Baxter’s 1973 paper. As an application, we prove the sum rule for the Bethe roots.

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

Access this article

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Baxter R.J.: Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain I. Ann. Phys. 76, 1–24 (1973)

    Article  ADS  MATH  Google Scholar 

  3. Baxter R.J.: Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain II. Ann. Phys. 76, 25–47 (1973)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Baxter R.J.: Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain III. Ann. Phys. 76, 48–71 (1973)

    Article  ADS  MATH  Google Scholar 

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

    MATH  Google Scholar 

  6. Bazhanov V.V., Frassek R., Łukowski T., Meneghelli C., Staudacher M.: Baxter Q-operators and representations of Yangians. Nuclear Phys. B 850, 148–174 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Bazhanov V.V., Lukyanov S.L., Zamolodchikov A.B.: Integrable structure of conformal field theory II, Q-operator and DDV equation. Commun. Math. Phys. 190, 247–278 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Bazhanov V.V., Lukyanov S.L., Zamolodchikov A.B.: Integrable structure of conformal field theory III, The Yang–Baxter relation. Commun. Math. Phys. 200, 297–324 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Bazhanov V.V., Stroganov Yu.G.: Chiral Potts model as a descendant of the six-vertex model. J. Stat. Phys. 59, 799–817 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Chicherin D., Derkachov S., Karakhanyan D., Kirschner R.: Baxter operators with deformed symmetry. Nuclear Phys. B 868, 652–683 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Chicherin, D., Derkachov, S.E., Spiridonov, V.P.: New elliptic solutions of the Yang–Baxter equation. arXiv:1412.3383

  12. Fabricius K.: A new Q-matrix in the eight-vertex model. J. Phys. A 40, 4075–4086 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Frenkel E., Hernandez D.: Baxter’s relations and spectra of quantum integrable models. Duke Math. J 164, 2407–2460 (2015)

    Article  MathSciNet  Google Scholar 

  14. Fabricius, K., McCoy, B.M.: New developments in the eight vertex model. J. Stat. Phys. 111, 323–337 (2003), ditto II, chains of odd length, J. Stat. Phys. 120, 37–70 (2005)

  15. Fabricius K., McCoy B.M.: The TQ equation of the eight-vertex model for complex elliptic roots of unity. J. Phys. A 40, 14893–14926 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Konno H.: The vertex-face correspondence and the elliptic 6j-symbols. Lett. Math. Phys. 72, 243–258 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Mangazeev V.V.: On the Yang–Baxter equation for the six-vertex model. Nuclear Phys. B 882, 70–96 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Mangazeev V.V.: Q-operators in the six-vertex model. Nuclear Phys. B 886, 166–184 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  19. Motegi, K.: On Baxter’s Q operator of the higher spin XXZ chain at the Razumov–Stroganov point. J. Math. Phys. 54, 063510 (2013)

  20. Mumford, D.: Tata lectures on Theta I. In: Progress in Mathematics, vol. 28. Birkhauser, Boston (1983)

  21. Roan, S.-S.: On Q-operators of XXZ Spin Chain of Higher Spin. arXiv:cond-mat/0702271

  22. Roan S.-S.: The Q-operator and functional relations of the eight-vertex model at root-of-unity \({\eta=\frac{2mK}{N}}\) for odd N. J. Phys. A40, 11019–11044 (2007)

    ADS  MathSciNet  Google Scholar 

  23. Rosengren H.: Sklyanin invariant integration. Int. Math. Res. Not. 60, 3207–3232 (2004)

    Article  MathSciNet  Google Scholar 

  24. Rosengren H.: An elementary approach to 6j-symbols (classical, quantum, rational, trigonometric, and elliptic). Ramanujan J. 13, 131–166 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sklyanin, E.K.: Some Algebraic structures connected with the Yang–Baxter equation. Funkt. Anal. Prilozh. 16-4, 27–34 (1982) (in Russian) (Funct. Anal. Appl. 16, 263–270, 1983) (English translation)

  26. Sklyanin, E.K.: Some algebraic structures connected with the Yang–Baxter equation. Representations of quantum algebras. Funkt. Anal Prilozh. 17-4, 34–48 (1983) (in Russian) (Funct. Anal. Appl., 17, 273–284, 1984) (English translation)

  27. Takebe T.: Generalized Bethe Ansatz with the general spin representations of the Sklyanin algebra. J. Phys. A 25, 1071–1083 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Takebe, T.: Bethe Ansatz for higher spin eight-vertex models. J. Phys. A 28, 6675–6706 (1995) (corrigendum J. Phys. A 29, 1563–1566 (1996))

  29. Takebe T.: Bethe ansatz for higher-spin XYZ models—low-lying excitations. J. Phys. A 29, 6961–6966 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Takebe T.: A system of difference equations with elliptic coefficients and Bethe vectors. Commun. Math. Phys. 183, 161–181 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Takhtajan, L.A., Faddeev, L.D.: The quantum method of the inverse problem and the Heisenberg XYZ model. Uspekhi Mat. Nauk. 34(5), 13–63 (1979) (in Russian) (Russ. Math. Surv. 34(5), 11–68, 1979) (English translation)

  32. Whittaker, E.T., Watson, G.N.: A course of modern analysis. In: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions, 4th edn. Cambridge University Press, New York (1927)

  33. Zabrodin A.: Commuting difference operators with elliptic coefficients from Baxter’s vacuum vectors. J. Phys. A 33, 3825–3850 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Takashi Takebe.

Additional information

Dedicated to Professor Evgeny Sklyanin on the occasion of his sixtieth birthday

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Takebe, T. Q-Operators for Higher Spin Eight Vertex Models with an Even Number of Sites. Lett Math Phys 106, 319–340 (2016). https://doi.org/10.1007/s11005-015-0813-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-015-0813-7

Mathematics Subject Classification

Keywords

Navigation