A New Numerical Method to Study Phase Transitions

  • J. M. Kosterlitz
  • J. Lee
  • E. Granato
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 72)


A powerful method of detecting first order transitions by numerical simulations of finite systems is presented. The method relies on simulations and the finite size scaling properties of free energy barriers between coexisting states. It is demonstrated that the first order transitions in d = 2, q = 5 and d = q = 3 Potts models are easily seen with modest computing time. The method can also be used to obtain quite accurate estimates of critical exponents by studying the barriers in the vicinity of a critical point. Some new results on exponents and conformal charge in frustrated XY models and a related coupled XY-Ising model in d = 2 are presented. These show that the transitions in these models are in new universality classes and that the conformal charge varies with a parameter.


Potts Model Ising Model Critical Exponent Order Transition Triangular Lattice 
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  1. 1.
    R.B. Potts, Proc. Camb. Phil. Soc. 48, 106 (1952); F.Y. Wu, Rev. Mod. Phys. 54, 235 (1982).CrossRefMATHADSMathSciNetGoogle Scholar
  2. 2.
    P. Peczak and D.P. Landau, Phys. Rev. B39, 11932 (1989).ADSGoogle Scholar
  3. 3.
    M.S.S. Challa, D.P. Landau and K. Binder, Phys. Rev. B34, 184 (1986).ADSGoogle Scholar
  4. 4.
    J.F. McCarthy, Phys. Rev. B41, 9530 (1990).ADSGoogle Scholar
  5. 5.
    M. Fukugita, II. Mino, M. Okawa and A. Ukawa, J. Phys. A23, L561, (1990).ADSGoogle Scholar
  6. 6.
    K. Binder, Rep. Prog. Phys. 50 783 (1987).CrossRefADSGoogle Scholar
  7. 7.
    K. Binder, Monte Carlo Methods in Statistical Physics, ed. K. Binder, Topics in Current Physics, (Springer, Berlin, Heidelberg and New York, 1979) vol. 7, p. 1.Google Scholar
  8. 8.
    D.P. Landau and K. Binder, Phys. Rev. B17, 2328 (1978).ADSGoogle Scholar
  9. 9.
    K. Binder and D.P. Landau, Phys. Rev. B21, 1941 (1980).ADSGoogle Scholar
  10. 10.
    K. Binder, Phys. Rev. Lett., 47, 693 (1981); Z. Phys. B43, 119 (1981).CrossRefADSGoogle Scholar
  11. 11.
    J. Lee and J.M. Kosterlitz, Phys. Rev. B43, 1268 (1991).ADSGoogle Scholar
  12. 12.
    J. Lee and J.M. Kosterlitz, Phys. Rev. Lett. 65 137 (1990).CrossRefMATHADSMathSciNetGoogle Scholar
  13. 13.
    M.N. Barber, “Phase Transitions and Critical Phenomena,”eds. C. Domb and J.L. Lebowitz (Academic, New York, 1983), Vol. 8, p. 145.Google Scholar
  14. 14.
    For recent reviews see “Finite Size Scaling and Numerical Simulations of Statistical Systems, ed. V. Privman (World Scientific, Singapore, 1990).Google Scholar
  15. 15.
    E. Eisenriegier and R. Tomaschitz, Phys. Rev. B35, 4876 (1987).ADSGoogle Scholar
  16. 16.
    A.M. Ferrenberg and R.H. Swendsen, Phys..Rev. Lett. 61, 2635 (1988)CrossRefADSGoogle Scholar
  17. 17.
    R.H. Swendsen and J-S Wang, Phys. Rev. Lett. 58, 86 (1987).CrossRefADSGoogle Scholar
  18. 18.
    J. Lee and J.M. Kosterlitz, Phys. Rev. B43, 3265, (1991).ADSGoogle Scholar
  19. 19.
    J. Lee, Ph.D. Thesis (unpublished).Google Scholar
  20. 20.
    S. Teitel and C. Jayaprakash, Phys. Rev. Lett. 51, 199 (1983); W.Y. Shih and D. Stroud, Phys. Rev. B32, 158 (1985).CrossRefGoogle Scholar
  21. 21.
    J. Villain, J. Phys. C10, 1717 (1977); ibid. 10, 4793 (1977).ADSGoogle Scholar
  22. 22.
    S. Miyashita and II. Shiba, J. Phys. Soc. Jpn. 53, 1145 (1984); D.H. Lee, J.D. Joannopoulos, J.W. Negele and D.P. Landau, Phys. Rev. B33, 450 (1986).CrossRefADSGoogle Scholar
  23. 23.
    T.C. Halsey, J. Phys. C18, 2437 (1986); S.E. Kurshonov and G.V. Uimin, J. Stat. Phys. 43, 1 (1986).ADSGoogle Scholar
  24. 24.
    B. Berge., H. Diep, A. Ghazali and P. Lallemand, Phys. Rev. B34, 3177 (1986).ADSGoogle Scholar
  25. 25.
    E. Granato, J.M. Kosterlitz, J. Lee and P.M. Nightingale, Phys. Rev. Lett. 66, (1991) in press.Google Scholar
  26. 26.
    J. Lee, E. Granato and J.M. Kosterlitz, unpublished.Google Scholar
  27. 27.
    J.M. Thijssen and II.J.F. Knops, Phys. Rev. B42, 4797 (1990).Google Scholar
  28. 28.
    J.M. Thijssen and H.J.F. Knops, Phys. Rev. B37, 7738 (1988).ADSGoogle Scholar
  29. 29.
    G. Grest, Phys. Rev. B39, 9267 (1989).ADSGoogle Scholar
  30. 30.
    A.B. Zamolodchikov, JETP Lett. 43, 732 (1986).ADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • J. M. Kosterlitz
    • 1
  • J. Lee
    • 2
  • E. Granato
    • 3
  1. 1.Department of PhysicsMcGill UniversityMontrealCanada
  2. 2.Materials Science DivisionArgonne National LaboratoryArgonneUSA
  3. 3.INPE12201-Sao José dos CamposBrazil

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