Skip to main content
SpringerLink
Log in

Finite size scaling analysis of ising model block distribution functions

  • Published:
Zeitschrift für Physik B Condensed Matter

Abstract

The distribution functionP L (s) of the local order parameters in finite blocks of linear dimensionL is studied for Ising lattices of dimensionalityd=2, 3 and 4. Apart from the case where the block is a subsystem of an infinite lattice, also the distribution in finite systems with free [P (f) L (s)] and periodic [P (p)(s) L ] boundary conditions is treated. Above the critical pointT c , these distributions tend for largeL towards the same gaussian distribution centered around zero block magnetization, while belowT c these distributions tend towards two gaussians centered at ±M, whereM is the spontaneous magnetization appearing in the infinite systems. However, belowT c the wings of the distribution at small |s| are distinctly nongaussian, reflecting two-phase coexistence. Hence the distribution functions can be used to obtain the interface tension between ordered phases.

At criticality, the distribution functions tend for largeL towards scaled universal forms, though dependent on the boundary conditions. These scaling functions are estimated from Monte Carlo simulations. For subsystem-blocks, good agreement with previous renormalization group work of Bruce is obtained.

As an application, it is shown that Monte Carlo studies of critical phenomena can be improved in several ways using these distribution functions:(i) standard estimates of order parameter, susceptibility, interface tension are improved(ii) T c can be estimated independent of critical exponent estimates(iii) A Monte Carlo “renormalization group” similar to Nightingale's phenomenological renormalization is proposed, which yields fairly accurate exponent estimates with rather moderate effort(iv) Information on coarse-grained hamiltonians can be gained, which is particularly interesting if the method is extended to more general Hamiltonians.

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. Kampen, N.G. van: Phys. Rev.135, A362 (1964).

    Google Scholar 

  2. Kadanoff, L.P.: Physics2, 263 (1966)

    Google Scholar 

  3. Langer, J.S.: Ann. Phys.65, 53 (1971); Physica73, 61 (1974)

    Google Scholar 

  4. Langer, J.S., Baron, M., Miller, H.D.: Phys. Rev. A11, 1417 (1975)

    Google Scholar 

  5. Billotet, C., Binder, K.: Z. Physik. B—Condensed Matter32, 195 (1979)

    Google Scholar 

  6. Kawasaki, K., Imaeda, T., Gunton, J.D.: In: Studies in Statistical Mechanics. Raveche, H. (ed.). Amsterdam: North-Holland 1980

    Google Scholar 

  7. Wilson, K.G.: Phys. Rev. B4, 3174 (1971); B4, 3184 (1971)

    Google Scholar 

  8. Fisher, M.E.: Rev. Mod. Phys.46, 587 (1974)

    Google Scholar 

  9. Domb, C., Green, M.S. (eds.): Phase Transitions and Critical Phenomena. Vol. 6. New York: Academic Press 1976

    Google Scholar 

  10. Le Guillou, J.C., Zinn-Justin, J.: Phys. Rev. B21, 3976 (1980)

    Google Scholar 

  11. Binder, K. (ed.): Monte Carlo Methods in Statistical Physics Berlin, Heidelberg, New York: Springer 1979

    Google Scholar 

  12. Bruce, A.D.: Preprint; see also Bruce, A.D., Schneider, T., Stoll, E.: Phys. Rev. Lett. 43, 1284 (1979)

  13. Nightingale, M.P.: Physica83A, 561 (1976); Proc. K. Ned. Acad. v. Wet. B82 (3), 235 (1979)

    Google Scholar 

  14. Sneddon, L.: J. Phys. C11, 2823 (1978); C12, 3051 (1979) dos Santos, R.R., Sneddon, L.: Preprint

    Google Scholar 

  15. Racz, Z.: Phys. Rev. B21, 4012 (1980)

    Google Scholar 

  16. Derrida, B., Vannimenus, J.: J. Phys. Lett.41, (1980)

  17. Derrida, B.: Preprint

  18. Nightingale, M.P., Blöte, H.W.J.: Physica104A, 352 (1980)

    Google Scholar 

  19. Schick, M., Kinzel, W.: (unpublished)

  20. Blöte, H.W.J., Nightingale, M.P., Derrida, B.: Preprint

  21. Ma, S.-K.: Phys. Rev. Lett.37, 461 (1976)

    Google Scholar 

  22. Friedman, Z., Felsteiner, J.: Phys. Rev. B15, 5317 (1977)

    Google Scholar 

  23. Reynolds, P.J., Stanley, H.E., Klein, W.: Phys. Rev. B21, 1223 (1980)

    Google Scholar 

  24. Herrmann, H.J., Stauffer, D., Eschbach, P.D.: Phys. Rev. B23, 422 (1981)

    Google Scholar 

  25. Swendsen, R.H.: Phys. Rev. Lett.42, 859 (1979); Phys. Rev. B20, 2080 (1979)

    Google Scholar 

  26. Blöte, H.W.J., Swendsen, R.H.: Phys. Rev. B20, 2077 (1979)

    Google Scholar 

  27. Blöte, H.W.J., Swendsen, R.H.: Phys. Rev. B22, 4481 (1980)

    Google Scholar 

  28. Blöte, H.W.J., Swendsen, R.H.: Phys. Rev. Lett.43, 737 (1979)

    Google Scholar 

  29. Swendsen, R.H., Krinsky, S.: Phys. Rev. Lett.43, 177 (1979)

    Google Scholar 

  30. Rebbi, C., Swendsen, R.H.: preprint; Novotny, M.A., Landau, D.P., Swendsen, R.H.: Preprint

  31. Landau, D.P., Swendsen, R.H.: Preprint

  32. Baumgärtner, A.: J. Phys. A13, L38 (1980)

    Google Scholar 

  33. Kremer, K., Baumgärtner, A., Binder, K.: Z. Phys. B—Condensed Matter40, 331 (1981)

    Google Scholar 

  34. Reoner, S., Reynolds, P.J.: Preprint

  35. Shenker, S., Tobochnik, J.: Phys. Rev. B22, 4462 (1980)

    Google Scholar 

  36. Landau, D.P.: Phys. Rev. B13, 2297 (1976); B14, 255 (1976)

    Google Scholar 

  37. For connecting probability theory and renormalization group ideas, see Jona-Lasinio, G.: I. Nuovo Cimento26B, 99 (1975)

    Google Scholar 

  38. Binder, K., Rauch, H.: Z. Phys.219, 201 (1969)

    Google Scholar 

  39. Schulman, L.S.: J. Phys. A13, 237 (1980)

    Google Scholar 

  40. Binder, K., Kalos, M.H.: J. Stat. Phys.22, 363 (1980)

    Google Scholar 

  41. Fisher, M.E.: In: Critical Phenomena Green, M.S. (ed.), p. 1. New York: Academic Press 1971

    Google Scholar 

  42. Suzuki, M.: Prog. Theor. Phys.58, 1142 (1977)

    Google Scholar 

  43. The correlation length ζ. which enters the second argument of\(\tilde P\) could differ from the standard definition of ζ by a constant of order unity. For simplicity this constant is here absorbed in the definition of ζ

  44. Patashinskii, A.Z.: Sov. Phys. JETP26, 1126 (1968)

    Google Scholar 

  45. Baker, G.A., Jr.: Phys. Rev. B15, 1552 (1977)

    Google Scholar 

  46. Baker, G.A., Jr., Kincaid, J.M.: J. Stat. Phys.24, 469 (1981)

    Google Scholar 

  47. Stell, G.: In: Critical Phenomena. Green, M.S. (ed.), p. 188. New York: Academic Press 1971

    Google Scholar 

  48. Fisher, M.E.: In: Proceedings of the Twenty-Fourth Nobel Symposium on Collective Properties of Physical Systems. Lundquist, B., Lundquist, S. (eds.). p. 16. New York: Academic Press 1973

    Google Scholar 

  49. Binder, K.: Thin Solid Films20, 367 (1974)

    Google Scholar 

  50. Baumgärtner, A., Binder, K.: J. Chem. Phys. (in press)

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

    Google Scholar 

  52. Because of the symmetryP L (s)=P L (−s) only data fors>0 are shown. The data in the figures arbitrarily are normalized to\(\int\limits_0^\infty {P_L (s)ds = 1} \) rather than 1/2

  53. Domb, C.: In: Phase Transitions and Critical Phenomena. Vol. 3. Domb, C., Green, M.S. (eds.), Vol. 3. New York: Academic Press 1974

    Google Scholar 

  54. Gaunt, D.S., Sykes, M.F., McKenzie, S.: J. Phys. A12, 871 (1979)

    Google Scholar 

  55. Mouritsen, O.G., Knak Jensen, S.J.: Phys. Rev. B19, 3663 (1979)

    Google Scholar 

  56. Brezin, E., Le Guillou, J.C., Zinn-Justin, J.: Phys. Rev. D8, 2418 (1973)

    Google Scholar 

  57. Binder, K.: (to be published)

  58. Kinzel, W.: Phys. Rev. B19, 4584 (1979)

    Google Scholar 

  59. Racz, Z., Rujan, P.: Z. Phys. B—Condensed Matter28, 287 (1977)

    Google Scholar 

  60. Muto, S., Oguchi, T., Ono, I.: J. Phys. A13, 1799 (1980)

    Google Scholar 

  61. Leeuwen, J.M.J. van: In: Phase Transitions and Critical Phenomena. Vol. 6. Domb, C., Green, M.S. (eds.), Vol. 6. New York: Academic Press 1976

    Google Scholar 

  62. Müller-Krumbhaar, H.: Z. Phys. B—Condensed Matter (1980)

  63. Nickel, B.G., Sharpe, B.: J. Phys. A12, 1819 (1979)

    Google Scholar 

  64. Gaunt, D.S., Sykes, M.F.: J. Phys. A12, L25 (1979)

    Google Scholar 

  65. Rehr, J.J.: J. Phys. A12, L179 (1979)

    Google Scholar 

  66. McKenzie, S.: J. Phys. A12, L185 (1979)

    Google Scholar 

  67. Zinn-Justin, J.: J. Phys. (Paris)40, 969 (1979)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Festkörperforschung, Kernforschungsanlage Jülich, Postfach 19 13, D-5170, Jülich 1, Germany

    K. Binder

Authors
  1. K. Binder
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Binder, K. Finite size scaling analysis of ising model block distribution functions. Z. Physik B - Condensed Matter 43, 119–140 (1981). https://doi.org/10.1007/BF01293604

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01293604

Keywords

Search

Navigation