Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Monte Carlo Simulations of Spin Systems

  • Chapter
Computational Physics

Abstract

This chapter gives a brief introduction to Monte Carlo simulations of classical O(n) spin systems such as the Ising (n = 1), XY (n = 2), and Heisenberg (n = 3) models. In the first part I discuss some aspects of the use of Monte Carlo algorithms to generate the raw data. Here special emphasis is placed on nonlocal cluster update algorithms which proved to be most efficient for this class of models. The second part is devoted to the data analysis at a continuous phase transition. For the example of the three-dimensional Heisenberg model it is shown how precise estimates of the transition temperature and the critical exponents can be extracted from the raw data. I conclude with a brief overview of recent results from similar high-precision studies of the Ising and XY models.

Software included on the accompanying diskette.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. T.H. Berlin and M. Kac, Phys. Rev. 86, 821 (1952)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. V.L. Berezinskii, Zh. Eksp. Teor. Fiz. 61, 1144 (1971) [Sov. Phys. JETP 34, 610 (1972)]

    Google Scholar 

  3. J.M. Kosterlitz and D.J. Thouless, J. Phys. C 6, 1181 (1973)

    Article  ADS  Google Scholar 

  4. J.M. Kosterlitz, J. Phys. C 7, 1046 (1974)

    Article  ADS  Google Scholar 

  5. H. Kleinert, Gauge Fields in Condensed Matter, Vol. I (World Scientific, Singapore 1989)

    MATH  Google Scholar 

  6. A. Patrascioiu and E. Seiler, Munich preprint MPI-PhT/95–73, hep-lat/9508014 (1995)

    Google Scholar 

  7. R.B. Potts, Proc. Camb. Phil. Soc. 48, 106 (1952)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. F.Y. Wu, Rev. Mod. Phys. 54, 235 (1982)

    Article  ADS  Google Scholar 

  9. F.Y. Wu, Rev. Mod. Phys. 55, 315(E) (1983)

    Article  ADS  Google Scholar 

  10. J.D. Gunton, M.S. Miguel, and P.S. Sahni, in Phase Transitions and Critical Phenomena, Vol. 8, eds. C. Domb and J.L. Lebowitz (Academic Press, New York 1983), p. 269

    Google Scholar 

  11. K. Binder, Rep. Prog. Phys. 50, 783 (1987)

    Article  ADS  Google Scholar 

  12. H.J. Herrmann, W. Janke, and F. Karsch, eds. Dynamics of First Order Phase Transitions (World Scientific, Singapore 1992)

    Google Scholar 

  13. W. Janke, in Computer Simulations in Condensed Matter Physics VII, eds. D.P. Landau, K.K. Mon, and H.-B. Schüttler (Springer, Berlin 1994), p. 29

    Google Scholar 

  14. H. Kleinert, Gauge Fields in Condensed Matter, Vol. II (World Scientific, Singapore 1989)

    Google Scholar 

  15. W. Janke and H. Kleinert, Phys. Rev. B 33, 6346 (1986)

    Article  ADS  Google Scholar 

  16. W. Janke and S. Kappler Nucl. Phys. B (Proc. Suppl.) 34, 674 (1994)

    Article  ADS  Google Scholar 

  17. W. Janke and S. Kappler, Phys. Lett. A 197, 227 (1995)

    Article  ADS  Google Scholar 

  18. W. Janke and S. Kappler, Nucl. Phys. B (Proc. Suppl.) 42, 770 (1995)

    Article  ADS  Google Scholar 

  19. W. Janke and S. Kappler, Europhys. Lett. 31, 345 (1995)

    Article  ADS  Google Scholar 

  20. W. Janke and S. Kappler, Mainz preprint KOMA-95–65 (September 1995), heplat/9509057, to appear in the proceedings of the conference Lattice’95, Melbourne, Australia, July 11–15, (1995)

    Google Scholar 

  21. S. Kappler, Ph.D. Thesis, Johannes Gutenberg-Universität Mainz (1995)

    Google Scholar 

  22. A.E. Ferdinand and M.E. Fisher, Phys. Rev. 185, 832 (1969)

    Article  ADS  Google Scholar 

  23. K. Binder, in Monte Carlo Methods in Statistical Physics, ed. K. Binder (Springer, Berlin 1979), p. 1

    Google Scholar 

  24. M.E. Barber, in Phase Transitions and Critical Phenomena, Vol. 8, eds. C. Domb and J.L. Lebowitz (Academic Press, New York 1983), p. 146

    Google Scholar 

  25. V. Privman, ed., Finite-Size Scaling and Numerical Simulations of Statistical Systems (World Scientific, Singapore 1990)

    Google Scholar 

  26. K. Binder, in Computational Methods in Field Theory, Schladming Lecture Notes, eds. H. Gausterer and C.B. Lang (Springer, Berlin 1992), p. 59

    Chapter  Google Scholar 

  27. M.E. Fisher and A.N. Berker, Phys. Rev. B 26, 2507 (1982)

    Article  ADS  Google Scholar 

  28. V. Privman and M.E. Fisher, J. Stat. Phys. 33, 385 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  29. K. Binder and D.P. Landau, Phys. Rev. B 30, 1477 (1984)

    Article  ADS  Google Scholar 

  30. M.S.S. Challa, D.P. Landau, and K. Binder, Phys. Rev. B 34, 1841 (1986)

    Article  ADS  Google Scholar 

  31. V. Privman and J. Rudnik, J. Stat. Phys. 60, 551 (1990)

    Article  MATH  ADS  Google Scholar 

  32. C. Borgs and R. Kotecky, J. Stat. Phys. 61, 79 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  33. C. Borgs and R. Kotecky, Phys. Rev. Lett. 68, 1734 (1992)

    Article  ADS  Google Scholar 

  34. J. Lee and J.M. Kosterlitz, Phys. Rev. Lett. 65, 137 (1990)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  35. J. Lee and J.M. Kosterlitz, Phys. Rev. B 43, 3265 (1991)

    Article  ADS  Google Scholar 

  36. C. Borgs, R. Kotecky, and S. Miracle-Solé, J. Stat. Phys. 62, 529 (1991)

    Article  ADS  Google Scholar 

  37. C. Borgs and W. Janke, Phys. Rev. Lett. 68, 1738 (1992)

    Article  ADS  Google Scholar 

  38. W. Janke, Phys. Rev. B 47, 14757 (1993)

    Article  ADS  Google Scholar 

  39. J.M. Hammersley and D.C. Handscomb, Monte Carlo Methods (London 1965)

    Google Scholar 

  40. K. Binder and D.W. Heermann, Monte Carlo Simulations in Statistical Physics: An Introduction (Springer, Berlin 1988)

    Google Scholar 

  41. D.W. Heermann, Computer Simulation Methods in Theoretical Physics, 2nd ed., (Springer, Berlin 1990)

    MATH  Google Scholar 

  42. K. Binder, ed. The Monte Carlo Method in Condensed Matter Physics (Springer, Berlin 1992)

    Google Scholar 

  43. N. Madras and A.D. Sokal, J. Stat. Phys. 50, 109 (1988)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  44. A.M. Ferrenberg, D.P. Landau, and K. Binder, J. Stat. Phys. 63, 867 (1991)

    Article  ADS  Google Scholar 

  45. W. Janke and T. Sauer, J. Stat. Phys. 78, 759 (1995)

    Article  MATH  ADS  Google Scholar 

  46. N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, J. Chem. Phys. 21, 1087 (1953)

    Article  ADS  Google Scholar 

  47. M. Creutz, Phys. Rev. D 36, 515 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  48. S.L. Adler, Phys. Rev. D 37, 458 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  49. H. Neuberger, Phys. Lett. B 207, 461 (1988)

    Article  ADS  Google Scholar 

  50. R. Gupta, J. DeLapp, G.G. Battrouni, G.C. Fox, C.F. Baillie, and J. Apostolakis, Phys. Rev. Lett. 61, 1996 (1988)

    Article  ADS  Google Scholar 

  51. J.D. Doll, R.D. Coalsen, and D.L. Freeman, Phys. Rev. Lett. 55, 1 (1985)

    Article  ADS  Google Scholar 

  52. J. Goodman and A.D. Sokal, Phys. Rev. Lett. 56, 1015 (1986)

    Article  ADS  Google Scholar 

  53. J. Goodman and A.D. Sokal, Phys. Rev. D 40, 2035 (1989)

    Article  ADS  Google Scholar 

  54. G. Mack, in Nonperturbative quantum field theory, Cargèse lectures 1987, ed. G.’t Hooft et al. (Plenum, New York 1988), p. 309

    Chapter  Google Scholar 

  55. D. Kandel, E. Domany, D. Ron, A. Brandt, and E. Loh, Jr., Phys. Rev. Lett. 60, 1591 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  56. D. Kandel, E. Domany, and A. Brandt, Phys. Rev. B 40, 330 (1989)

    Article  ADS  Google Scholar 

  57. G. Mack and S. Meyer, Nucl. Phys. B (Proc. Suppl.) 17, 293 (1990)

    Article  ADS  Google Scholar 

  58. A.D. Sokal, Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms, Cours de Troisième Cycle de la Physique en Suisse Romande, Lausanne (1989)

    Google Scholar 

  59. A.D. Sokal, Bosonic Algorithms, in Quantum Fields on the Computer, ed. M. Creutz (World Scientific, Singapore 1992), p. 211

    Google Scholar 

  60. R.H. Swendsen and J.-S. Wang, Phys. Rev. Lett. 58, 86 (1987)

    Article  ADS  Google Scholar 

  61. U. Wolff, Phys. Rev. Lett. 62, 361 (1989)

    Article  ADS  Google Scholar 

  62. P.W. Kasteleyn and C.M. Fortuin, J. Phys. Soc. Japan 26 (Suppl.), 11 (1969)

    ADS  Google Scholar 

  63. C.M. Fortuin and P.W. Kasteleyn, Physica 57, 536 (1972)

    Article  ADS  MathSciNet  Google Scholar 

  64. C.M. Fortuin, Physica 58, 393 (1972)

    Google Scholar 

  65. C.M. Fortuin, Physica 59, 545 (1972)

    Google Scholar 

  66. U. Wolff, Nucl. Phys. B 322, 759 (1989)

    Article  ADS  Google Scholar 

  67. U. Wolff, Nucl. Phys. B 334, 581 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  68. M. Hasenbusch, Nucl. Phys. B 333, 581 (1990)

    Article  ADS  Google Scholar 

  69. C.F. Baillie, Int. J. Mod. Phys. C 1, 91 (1990)

    Article  ADS  Google Scholar 

  70. R.H. Swendsen, J.-S. Wang, and A.M. Ferrenberg, in The Monte Carlo Method in Condensed Matter Physics, ed. K. Binder (Springer, Berlin 1992), p. 75

    Google Scholar 

  71. U. Wolff, Phys. Lett. A 228, 379 (1989)

    Google Scholar 

  72. D.W. Heermann and A.N Burkitt, Physica A 162, 210 (1990)

    Article  ADS  Google Scholar 

  73. P. Tamayo, Physica A 201, 543 (1993)

    Article  ADS  Google Scholar 

  74. N. Ito and G.A. Kohring, Physica A 201, 547 (1993)

    Article  ADS  Google Scholar 

  75. M. Hasenbusch and S. Meyer, Phys. Lett. B 241, 238 (1990)

    Article  ADS  Google Scholar 

  76. W. Janke, Phys. Lett. A 148, 306 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  77. C. Holm and W. Janke, Phys. Rev. B 48, 936 (1993)

    Article  ADS  Google Scholar 

  78. K. Binder, Z. Phys. B 43, 119 (1981)

    Article  ADS  Google Scholar 

  79. K. Binder, Phys. Rev. A 25, 1699 (1982)

    Article  ADS  Google Scholar 

  80. A. Billoire, R. Lacaze, A. Morel, S. Gupta, A. Irbäck, and B. Petersson, Nucl. Phys. B 358, 231 (1991)

    Article  ADS  Google Scholar 

  81. B.A. Berg and T. Neuhaus, Phys. Lett. B 267, 249 (1991)

    Article  ADS  Google Scholar 

  82. B.A. Berg and T. Neuhaus, Phys. Rev. Lett. 68, 9 (1992)

    Article  ADS  Google Scholar 

  83. W. Janke, B.A. Berg, and M. Katoot, Nucl. Phys. B 382, 649 (1992)

    Article  ADS  Google Scholar 

  84. K. Rummukainen, Nucl. Phys. B 390, 621 (1993)

    Article  ADS  Google Scholar 

  85. W. Janke and T. Sauer, Phys. Rev. E 49, 3475 (1994)

    Article  ADS  Google Scholar 

  86. T. Sauer, Ph.D. Thesis, Freie Universität Berlin (1994)

    Google Scholar 

  87. W. Janke and S. Kappler, Nucl. Phys. B (Proc. Suppl.) 42, 876 (1995)

    Article  ADS  Google Scholar 

  88. W. Janke and S. Kappler, Phys. Rev. Lett. 74, 212 (1995)

    Article  ADS  Google Scholar 

  89. A.M. Ferrenberg and A.M. Swendsen, Phys. Rev. Lett. 63, 1658(E) (1989)

    Article  ADS  Google Scholar 

  90. A.M. Ferrenberg and A.M. Swendsen, Phys. Rev. Lett. 61, 2635 (1988)

    Article  ADS  Google Scholar 

  91. N.A. Alves, B.A. Berg, and S. Sanielevici, Nucl. Phys. B 376, 218 (1992)

    Article  ADS  Google Scholar 

  92. R.G. Miller, Biometrika 61, 1 (1974)

    MATH  MathSciNet  Google Scholar 

  93. B. Efron, The Jackknife, the Bootstrap and Other Resampling Plans (SIAM, Philadelphia 1982)

    Book  Google Scholar 

  94. A.M. Ferrenberg and A.M. Swendsen, Phys. Rev. Lett. 63, 1195 (1989)

    Article  ADS  Google Scholar 

  95. P. Peczak and D.P. Landau, Bull. Am. Phys. Soc. 35, 255 (1990)

    Google Scholar 

  96. P. Peczak and D.P. Landau, J. Appl. Phys. 67, 5427 (1990)

    Article  ADS  Google Scholar 

  97. P. Peczak and D.P. Landau, Phys. Rev. B 47, 14260 (1993)

    Article  ADS  Google Scholar 

  98. P. Peczak, A.M. Ferrenberg, and D.P. Landau, Phys. Rev. B 43, 6087 (1991)

    Article  ADS  Google Scholar 

  99. C. Holm and W. Janke, Phys. Lett. A 173, 8 (1993)

    Article  ADS  Google Scholar 

  100. C. Holm and W. Janke, J. Phys. A 27, 2553 (1994)

    Article  MATH  ADS  Google Scholar 

  101. E. Brézin and J. Zinn-Justin, Nucl. Phys. B 257 [FS14], 867 (1985)

    Article  ADS  Google Scholar 

  102. J.C. Le Guillou and J. Zinn-Justin, Phys. Rev. Lett. 39, 95 (1977)

    Article  ADS  Google Scholar 

  103. J.C. Le Guillou and J. Zinn-Justin, Phys. Rev. B 21, 3976 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  104. J.C. Le Guillou and J. Zinn-Justin, J. Physique Lett. 46, L137 (1985)

    Article  Google Scholar 

  105. I. Dimitrovic, P. Hasenfratz, J. Nager, and F. Niedermayer, Nucl. Phys. B 350, 893 (1991)

    Article  ADS  Google Scholar 

  106. K. Chen, A.M. Ferrenberg, and D.P. Landau, Phys. Rev. B 48, 3249 (1993)

    Article  ADS  Google Scholar 

  107. J. Adler, C. Holm, and W. Janke, Physica A 201, 581 (1993)

    Article  ADS  Google Scholar 

  108. P. Butera and M. Comi, Phys. Rev. B 52, 6185 (1995)

    Article  ADS  Google Scholar 

  109. C.F. Baillie, R. Gupta, K.A. Hawick, and G.S. Pawley, Phys. Rev. B 45, 10438 (1992)

    Article  ADS  Google Scholar 

  110. A.M. Ferrenberg and D.P. Landau, Phys. Rev. B 44, 5081 (1991)

    Article  ADS  Google Scholar 

  111. H.W.J. Blöte, E. Luijten, and J.R. Heringa, J. Phys. A 28, 6289 (1995)

    Article  MATH  ADS  Google Scholar 

  112. J. Adler, J. Phys. A 16, 3585 (1983)

    Article  ADS  Google Scholar 

  113. A. Rosengren, J. Phys. A 19, 1709 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  114. L.S. Goldner and G. Ahlers, Phys. Rev. B 45, 13129 (1992)

    Article  ADS  Google Scholar 

  115. A.P. Gottlob and M. Hasenbusch, Physica A 201, 593 (1993)

    Article  ADS  Google Scholar 

  116. P. Butera, M. Comi, and A.J. Guttmann, Phys. Rev. B 48, 13987 (1993)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Physik, Johannes Gutenberg-Universität, D-55099, Mainz, Germany

    Wolfhard Janke

Authors
  1. Wolfhard Janke
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Physik, Technische Universität Chemnitz-Zwickau, D-09107, Chemnitz, Germany

    Karl Heinz Hoffmann  & Michael Schreiber  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Janke, W. (1996). Monte Carlo Simulations of Spin Systems. In: Hoffmann, K.H., Schreiber, M. (eds) Computational Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-85238-1_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-85238-1_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-85240-4

  • Online ISBN: 978-3-642-85238-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation