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Corner transfer matrices of the chiral Potts model. II. The triangular lattice

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Abstract

We consider a two-dimensional edge-interaction model satisfying the star-triangle relations. For the triangular lattice, the corner transfer matrices are functions of three rapidities: we show that they possess various factorization properties and satisfy certain equations. We indicate how these equations can be solved for the Ising model. We then consider the three-state chiral Potts model and obtain low-temperature solutions to the equations. The conjectured formula for the order parameter (the spontaneous magnetization) is verified to one more order in a series expansion.

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References

  1. 1.

    L. Dolan and M. Grady,Phys. Rev. D 25:1587–1604 (1982).

  2. 2.

    S. Howes, L. P. Kadanoff, and M. den Nijs,Nucl. Phys. B 215[FS7]:169–208 (1983).

  3. 3.

    G. von Gehlen and V. Rittenberg,Nucl. Phys. B 257[FS14]:351–370 (1985).

  4. 4.

    H. Au-Yang, B. M. McCoy, J. H. H. Perk, S. Tang, and M.-L. Yan,Phys. Lett. A 123:219–223 (1987).

  5. 5.

    J. H. H. Perk, inTheta Functions Bowdoin 1987 (Proceedings Symposia in Pure Mathematics, Vol.49, Part 1), L.Ehrenpreis and R. C. Gunning, eds. (American Mathematical Society, 1989), pp. 341–354.

  6. 6.

    B. M. McCoy, J. H. H. Perk, and S. Tang,Phys. Lett. A 125:9–14 (1987).

  7. 7.

    H. Au-Yang, B. M. McCoy, J. H. H. Perk, and S. Tang, inAlgebraic Analysis, Vol. 1, M. Kashiwara and T. Kawai, eds. (Academic Press, New York, 1988, pp. 29–39.

  8. 8.

    R. J. Baxter, J. H. H. Perk, and H. Au-Yang,Phys. Lett. A 128:138–142 (1988).

  9. 9.

    H. Au-Yang and J. H. H. Perk, inAdvanced Studies in Pure Mathematics, K. Aomoto and T. Oda, eds. (Academic/Kinokumiya, 1989), pp. 57–94.

  10. 10.

    V. B. Matveev and A. O. Smirnov,Lett. Math. Phys. 19:179–185 (1990).

  11. 11.

    V. A. Fateev and A. B. Zamolodchikov,Phys. Lett. A 92:37–39 (1982).

  12. 12.

    R. J. Baxter,Exactly Solved Models in Statistical Mechanics (Academic Press, New York, 1982).

  13. 13.

    R. J. Baxter,J. Stat. Phys. 52:639–667 (1988).

  14. 14.

    G. Albertini, B. M. McCoy, and J. H. H. Perk,Phys. Lett. A 135:159–166 (1989).

  15. 15.

    G. Albertini, B. M. McCoy, J. H. H. Perk, and S. Tang,Nucl. Phys. B 314:741–763 (1989).

  16. 16.

    G. Albertini, B. M. McCoy, and J. H. H. Perk,Phys. Lett. A 139:204–212 (1989).

  17. 17.

    V. V. Bazhanov and Yu. G. Stroganov,J. Stat. Phys. 59:799–817 (1990).

  18. 18.

    R. J. Baxter, V. V. Bazhanov, and J. H. H. Perk,Int. J. Mod. Phys. B 4:803–870 (1990).

  19. 19.

    V. O. Tarasov,Phys. Lett. A 147:487–490 (1990).

  20. 20.

    V. B. Matveev and A. O. Smirnov, inInverse Methods in Action, P. C. Sabatier, ed. (Springer-Verlag, Berlin, 1990), pp. 560–569.

  21. 21.

    R. J. Baxter,Phys. Lett. A 146:110–114 (1990).

  22. 22.

    R. J. Baxter, inProceedings Fourth Asia-Pacific Physics Conference, S. H. Ahn, II-T. Cheon, S. H. Choh, and C. Lee, eds. (World Scientific, Singapore, 1991), Vol.1, pp. 42–57.

  23. 23.

    B. M. McCoy and S. Roan,Phys. Lett. A 150:347–354 (1990).

  24. 24.

    G. Albertini, S. Dasmahapatra, and B. M. McCoy,J. Mod. Phys. A 7 (Supple. 1A):1–54 (1992).

  25. 25.

    R. J. Baxter,Phys. Lett. A 133:185–189 (1988).

  26. 26.

    R. J. Baxter,J. Stat. Phys. 57:1–39(1989).

  27. 27.

    R. J. Baxter,Phys. Lett. A 140:155–157 (1989).

  28. 28.

    V. O. Tarasov, Cyclic monodromy matrices for the R-matrix of the six-vertex model and the chiral Potts model with fixed-spin boundary conditions, RIMS-774, Kyoto (1991).

  29. 29.

    G. Albertini, B. M. McCoy, and J. H. H. Perk,Adv. Studies Pure Math. 19:1–55 (1989).

  30. 30.

    G. Albertini and B. M. McCoy,Nucl. Phys. B 350:745–788 (1991).

  31. 31.

    B. M. McCoy, inICM-90 Satellite Conference Proceedings Special Functions, Mc. Kashiwara and T. Miwa, eds. (Springer-Verlag, Berlin, 1991), pp. 245–259.

  32. 32.

    H. Itoyama, B. M. McCoy, and J. H. H. Perk,Int. J. Mod. Phys. b 4:995–1001 (1990).

  33. 33.

    V. V. Bazhanov, R. M. Kashaev, V. V. Mangazeev, and Yu. G. Stroganov,Commun. Math. Phys. 138:393–408 (1991).

  34. 34.

    V. V. Bazhanov and R. J. Baxter, New solvable lattice models in three dimensions,J. Stat. Phys. (1992).

  35. 35.

    A. B. Zamolodchikov,Commun. Math. Phys. 79:489–505 (1981).

  36. 36.

    R. J. Baxter,Commun. Math. Phys. 88:185–205 (1983).

  37. 37.

    R. J. Baxter,Physica D 18:3421–347 (1986).

  38. 38.

    R. J. Baxter,J. Stat. Phys. 63:433–453 (1991).

  39. 39.

    R. J. Baxter,J. Stat. Phys. 19:461–478 (1978).

  40. 40.

    R. J. Baxter,Physica A 106:18–27 (1981).

  41. 41.

    E. Date, M. Jimbo, T. Miwa, and M. Okado, inTheta Functions Bowdoin 1986 (Proceedings Symposia in Pure Mathematics, Vol. 49, Part 1), L. Ehrenpreis and R. C. Gunning, eds. (American Mathematical Society, 1989), pp. 295–331.

  42. 42.

    M. Henkel and J. Lacki,Phys. Lett. A 138:105–109 (1989).

  43. 43.

    R. J. Baxter, inProceedings of the International Congress of Mathematicians, Vol. 2, I. Satake, ed. (Springer-Verlag, Berlin, 1991), pp. 1305–1317.

  44. 44.

    R. J. Baxter and S. K. Tsang,J. Phys. A 13:1023–1030 (1980).

  45. 45.

    R. J. Baxter,J. Math. Phys. 9:650–654 (1968).

  46. 46.

    R. J. Baxter and I. G. Enting,J. Stat. Phys. 21:103–123 (1979).

  47. 47.

    R. J. Baxter, I. G. Enting, and S. K. Tsang,J. Stat. Phys. 22:465–489 (1980).

  48. 48.

    R. J. Baxter,Phil. Trans. R. Soc. Lond. 289:315–346 (1978).

  49. 49.

    H. B. Thacker,Physica D 18:348–359 (1986).

  50. 50.

    R. J. Baxter,J. Stat. Phys. 28:1–41 (1982).

  51. 51.

    I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series and Products (Academic Press, New York, 1965).

  52. 52.

    L. Onsager,Phys. Rev. 65:117–149 (1944).

  53. 53.

    Yu. G. Stroganov,Phys. Lett. 74A:116–118 (1979).

  54. 54.

    R. J. Baxter,Ann. Phys. 70:193–228 (1972).

  55. 55.

    L. Onsager,Nuovo Cimento (Suppl.)6:261 (1949).

  56. 56.

    C. N. Yang,Phys. Rev. 85:808–816 (1962).

  57. 57.

    R. J. Baxter and S. B. Kelland,J. Phys. C 7:L403-L406 (1974).

  58. 58.

    B. Kaufman,Phys. Rev. 49:1232–1243 (1949).

  59. 59.

    S. K. Tsang,J. Stat. Phys. 20:95–114 (1979).

  60. 60.

    R. J. Baxter,Ann. Israel Phys. Soc. 2(1):37–47 (1978).

  61. 61.

    R. J. Baxter, “Elliptic parametrization of the three-state chiral Potts model,” preprint NI92005 of the Isaac Newton Institute for Mathematical Sciences, University of Cambridge.

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Baxter, R.J. Corner transfer matrices of the chiral Potts model. II. The triangular lattice. J Stat Phys 70, 535–582 (1993). https://doi.org/10.1007/BF01053584

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Key words

  • Statistical mechanics
  • lattice models
  • chiral Potts model
  • corner transfer matrices