Skip to main content
SpringerLink
Log in

Cluster Monte Carlo and dynamical scaling for long-range interactions

  • Regular Article
  • Published:
The European Physical Journal Special Topics Aims and scope Submit manuscript

Abstract

Many spin systems affected by critical slowing down can be efficiently simulated using cluster algorithms. Where such systems have long-range interactions, suitable formulations can additionally bring down the computational effort for each update from O(N 2) to O(N ln N) or even O(N), thus promising an even more dramatic computational speed-up. Here, we review the available algorithms and propose a new and particularly efficient single-cluster variant. The efficiency and dynamical scaling of the available algorithms are investigated for the Ising model with power-law decaying interactions.

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. N. Kawashima, H. Rieger, in Frustrated Spin Systems, edited by H.T. Diep (World Scientific, Singapore, 2005), Chap. 9, p. 491

  2. N.G. Fytas, V. Martín-Mayor, M. Picco, N. Sourlas, Phys. Rev. Lett. 116, 227201 (2016)

    Article  ADS  Google Scholar 

  3. M. Wittmann, A.P. Young, Phys. Rev. E 90, 062137 (2014)

    Article  ADS  Google Scholar 

  4. E. Flores-Sola, B. Berche, R. Kenna, M. Weigel, Phys. Rev. Lett. 116, 115701 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  5. M.E. Fisher, Rev. Mod. Phys. 46, 597 (1974)

    Article  ADS  Google Scholar 

  6. D.C. Rapaport, The Art of Molecular Dynamics Simulation (Cambridge University Press, Cambridge, 2004)

  7. K. Binder, D.P. Landau, A Guide to Monte Carlo Simulations in Statistical Physics, 4th edn. (Cambridge University Press, Cambridge, 2015)

  8. N. Metropolis et al., J. Chem. Phys. 21, 1087 (1953)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  11. E. Luijten, Lect. Notes Phys. 703, 13 (2006)

    Article  ADS  Google Scholar 

  12. Y. Deng, T.M. Garoni, A.D. Sokal, Phys. Rev. Lett. 98, 230602 (2007)

    Article  ADS  Google Scholar 

  13. E.M. Elçi, M. Weigel, Phys. Rev. E 88, 033303 (2013)

    Article  ADS  Google Scholar 

  14. C. Lacroix, P. Mendels, F. Mila, Introduction to Frustrated Magnetism: Materials, Experiments, Theory (Springer, Berlin, 2011), Vol. 164

  15. J.W. Britton et al., Nature 484, 489 (2012)

    Article  ADS  Google Scholar 

  16. M.E. Fisher, S.K. Ma, B.G. Nickel, Phys. Rev. Lett. 29, 917 (1972)

    Article  ADS  Google Scholar 

  17. F.J. Dyson, Commun. Math. Phys. 12, 91 (1969)

    Article  ADS  Google Scholar 

  18. F.J. Dyson, Commun. Math. Phys. 21, 269 (1971)

    Article  ADS  Google Scholar 

  19. H.-J. Xu, B. Bergersen, Z. Rácz, Phys. Rev. E 47, 1520 (1993)

    Article  ADS  Google Scholar 

  20. J.F. Nagle, J.C. Bonner, J. Phys. C 3, 352 (1970)

    Article  ADS  Google Scholar 

  21. Z. Glumac, K. Uzelac, J. Phys. A 22, 4439 (1989)

    Article  ADS  Google Scholar 

  22. E. Luijten, H.W. Blöte, Int. J. Mod. Phys. C 6, 359 (1995)

    Article  ADS  Google Scholar 

  23. K. Fukui, S. Todo, J. Comp. Phys. 228, 2629 (2009)

    Article  ADS  Google Scholar 

  24. W. Janke, in Computational Physics, edited by K.H. Hoffmann, M. Schreiber (Springer, Berlin, 1996), pp. 10–43

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

    Article  ADS  MathSciNet  Google Scholar 

  26. R.G. Edwards, A.D. Sokal, Phys. Rev. D 38, 2009 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  27. M. Weigel, Phys. Rev. E 84, 036709 (2011)

    Article  ADS  Google Scholar 

  28. E. Luijten, H.W.J. Blöte, Phys. Rev. B 56, 8945 (1997)

    Article  ADS  Google Scholar 

  29. D.E. Knuth, The Art of Computer Programming, Volume 2: Seminumerical Algorithms, 3rd edn. (Addison-Wesley, Upper Saddle River, NJ, 1997)

  30. J.E. Gentle, Random number generation and Monte Carlo methods, 2nd edn. (Springer, Berlin, 2003)

  31. R.E. Tarjan, J. ACM 22, 215 (1975)

    Article  Google Scholar 

  32. M.E.J. Newman, R.M. Ziff, Phys. Rev. E 64, 016706 (2001)

    Article  ADS  Google Scholar 

  33. W. Janke, in Proceedings of the Euro Winter School “Quantum Simulations of Complex Many-Body Systems: From Theory to Algorithms”, Vol. 10 of NIC Series, edited by J. Grotendorst, D. Marx, A. Muramatsu (John von Neumann Institute for Computing, Jülich, 2002), pp. 423–445

  34. N. Persky, R. Ben-Av, I. Kanter, E. Domany, Phys. Rev. E 54, 2351 (1996)

    Article  ADS  Google Scholar 

  35. Y. Deng et al., Phys. Rev. Lett. 99, 055701 (2007)

    Article  ADS  Google Scholar 

  36. U. Wolff, Phys. Lett. B 228, 379 (1989)

    Article  ADS  Google Scholar 

  37. T.S. Ray, P. Tamayo, W. Klein, Phys. Rev. A 39, 5949 (1989)

    Article  ADS  Google Scholar 

  38. S.A. Cannas, D.A. Stariolo, F.A. Tamarit, Physica A 294, 362 (2001)

    Article  ADS  Google Scholar 

  39. B. Berche, R. Kenna, J.C. Walter, Nucl. Phys. B 865, 115 (2012)

    Article  ADS  Google Scholar 

  40. E.J. Flores-Sola, B. Berche, R. Kenna, M. Weigel, Eur. Phys. J. B 88, 1 (2015)

    Article  Google Scholar 

  41. E. Luijten, Ph.D. thesis, Delft University of Technology, 1997

  42. M. Sweeny, Phys. Rev. B 27, 4445 (1983)

    Article  ADS  Google Scholar 

  43. E.M. Elçi, Ph.D. thesis, Coventry University, 2015

  44. M.C. Angelini, G. Parisi, F. Ricci-Tersenghi, Phys. Rev. E 89, 062120 (2014)

    Article  ADS  Google Scholar 

  45. E. Flores-Sola, Ph.D. thesis, Coventry University, Coventry, 2016

  46. F. Beyer, M. Weigel, M.A. Moore, Phys. Rev. B 86, 014431 (2012)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, United Kingdom

    Emilio Flores-Sola, Martin Weigel & Ralph Kenna

  2. Institut Jean Lamour, CNRS/UMR 7198, Groupe de Physique Statistique, Université de Lorraine, BP 70239, F-54506, Vandœuvre-les-Nancy Cedex, France

    Emilio Flores-Sola & Bertrand Berche

  3. Doctoral College for the Statistical Physics of Complex Systems, Leipzig-Lorraine-Lviv-Coventry (L4), 04009, Leipzig, Germany

    Emilio Flores-Sola, Martin Weigel, Ralph Kenna & Bertrand Berche

Authors
  1. Emilio Flores-Sola
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Weigel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ralph Kenna
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Bertrand Berche
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Weigel.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Flores-Sola, E., Weigel, M., Kenna, R. et al. Cluster Monte Carlo and dynamical scaling for long-range interactions. Eur. Phys. J. Spec. Top. 226, 581–594 (2017). https://doi.org/10.1140/epjst/e2016-60338-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjst/e2016-60338-3

Search

Navigation