Skip to main content

A symbolic algorithm for finding exactly soluble statistical mechanical models

Abstract

In general it is a very difficult task to find statistical mechanical models which satisfy the Yang-Baxter equations and thus are completely integrable. We propose a new approach leading to a (overdetermined) set oflinear equations. The formalism is applied to the Ising and the Ashkin-Teller models, which are both self-duals in two dimensions. Preliminary results for a symbolic algebra manipulation program is given, which would derive the relevant set of equations for an arbitrary internal spin symmetry group.

This is a preview of subscription content, access via your institution.

References

  1. For an example see G. Györgyi and P. Ruján, J. Phys., C17, 4207, 1984 and references therein.

    Google Scholar 

  2. For example the exact solution of the hard hexagon model (R. J. Baxter, J. Phys., A13, L61, 1980, and J. Stat. Phys.,26, 427, 1981) confirmed earlier conjectures regarding the critical exponents of the 3-state Potts model by M. P. M. den Nijs, J. Phys., A12, 1857, 1979.

  3. R. J. Baxter, Ann. Phys. (N. Y.),76, 1, 1973; ibid.76, 25, 1973, ibid. Ann. Phys. (N. Y.),76, 48, 1973.

    Article  ADS  Google Scholar 

  4. G. E. Andrews, R. J. Baxter and P. J. Forrester, J. Stat. Phys.,35, 193, 1984.

    MATH  Article  ADS  MathSciNet  Google Scholar 

  5. P. Ruján, in Lecture Notes in Physics,226, p. 286, ed. N. Sanchez, Springer, New York, 1985.

    Google Scholar 

  6. R. J. Baxter, Exactly Soved Models in Statistical Mechanics, Academic, London, 1982; P. W. Kasteleyn, in Fundamental Problems in Statistical Mechanics, Vol. 3, p. 103, ed. E. D. G. Cohen, North-Holland, Amsterdam, 1975.

    Google Scholar 

  7. L. D. Faddeev, Sov. Sci. Rev. Section C (Math. Phys.), Vol.1, ed. P. Novikov, 1981; L. A. Takhtadzhand and L. D. Faddeev, Russian Math. Surveys,34, 11, 1979.

  8. H. B. Thacker, Rev. Mod. Phys.,53, 253, 1981.

    Article  ADS  MathSciNet  Google Scholar 

  9. S. K. Pokrovsky and Yu. A. Bashilov, Comm. Math. Phys.,84, 103, 1982.

    Article  ADS  MathSciNet  Google Scholar 

  10. L. Mittag and M. J. Stephen, J. Math. Phys.,12, 441, 1971.

    Article  ADS  Google Scholar 

  11. S. Elitzur, R. B. Pearson and J. Shigemitsu, Phys. Rev. D19, 3698, 1979.

    ADS  Google Scholar 

  12. J. L. Cardy, J. Phys., A13, 1507, 1980; P. Ruján, G. O. Williams, H. L. Frisch and G. Forgács, Phys. Rev., B23, 1362, 1981.

    ADS  MathSciNet  Google Scholar 

  13. L. Onsager, Phys. Rev.,65, 117, 1944.

    MATH  Article  ADS  MathSciNet  Google Scholar 

  14. R. J. Baxter, J. Phys. C6, L445, 1973.

    ADS  Google Scholar 

  15. R. J. Baxter, Proc. Roy. Soc. London, A383, 43, 1982.

    Article  ADS  MathSciNet  Google Scholar 

  16. H. N. V. Temperley and E. H. Lieb, Proc. Roy. Soc. London, A322, 251, 1971; R. J. Baxter, S. B. Kelland and F. Y. Wu, J. Phys. A9, 397, 1976.

    MATH  Article  ADS  MathSciNet  Google Scholar 

  17. F. Wegner, J. Phys., C5, L131, 1972.

    ADS  Google Scholar 

  18. J. M. Maillard, P. Ruján and T. T. Truong, J. Phys., A18, 3399, 1985.

    ADS  MathSciNet  Google Scholar 

  19. V. A. Fateev and A. B. Zamolodchikov, Phys. Lett., A92, 37, 1982.

    Article  MathSciNet  Google Scholar 

  20. A. A. Belavin, Nucl. Phys., B180(FS2), 189, 1981.

    Article  ADS  MathSciNet  Google Scholar 

  21. R. J. Baxter, Comm. Math. Phys.,88, 185, 1983.

    Article  ADS  MathSciNet  Google Scholar 

  22. M. Lüscher, Nucl. Phys., B117, 475, 1976.

    Article  ADS  Google Scholar 

  23. E. Fradkin and L. Susskind, Phys. Rev., D17, 2637, 1978.

    ADS  MathSciNet  Google Scholar 

  24. M. J. Stephen and L. Mittag, J. Math. Phys.,13, 1944, 1972; R. Savit, Rev. Mod. Phys.,52, 453, 1980.

    Article  ADS  MathSciNet  Google Scholar 

  25. L. Doland and M. Grady, Phys. Rev., D25, 1587, 1982.

    Article  ADS  MathSciNet  Google Scholar 

  26. G. V. Gehlen and V. Rittenberg, to be published.

  27. M. Suzuki, Progr. Theor. Phys.,46, 1377, 1971.

    ADS  Google Scholar 

  28. P. Ruján, J. Stat. Phys.,29, 231, 1982; J. Kurmann, H. Thomas and G. Müller, Physica,112A, 235, 1982.

    Article  ADS  Google Scholar 

  29. see Ref. [24a] ; R. Savit, Rev. Mod. Phys.,52, 453, 1980.

    Article  ADS  MathSciNet  Google Scholar 

  30. J. Ashkin and E. Teller, Phys. Rev.,64, 178, 1943.

    Article  ADS  Google Scholar 

  31. C. Fan, Phys. Lett., A39, 136, 1972.

    Article  Google Scholar 

  32. “Reduce User’s Manual” by A. C. Hearn, Version 3.1, Rand Publications, 1984 and Seven Reduce Interactive Lessons, by D. R. Stoutmayer.

Download references

Author information

Authors and Affiliations

Authors

Additional information

Dedicated to Prof. K. Nagy on his 60th birthday

Address: Institut für Festkörperforschung der Kernforschungsanlage Jülich, D-5170 Jülich, BRD, Postfach 1913

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ruján, P. A symbolic algorithm for finding exactly soluble statistical mechanical models. Acta Physica Hungarica 62, 287–306 (1987). https://doi.org/10.1007/BF03155980

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03155980

Keywords

  • Ising Model
  • Transfer Matrix
  • Matching Condition
  • Statistical Mechanical Model
  • Symbolic Algorithm