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Scaling and universality in the phase diagram of the 2D Blume-Capel model

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Abstract

We review the pertinent features of the phase diagram of the zero-field Blume-Capel model, focusing on the aspects of transition order, finite-size scaling and universality. In particular, we employ a range of Monte Carlo simulation methods to study the 2D spin-1 Blume-Capel model on the square lattice to investigate the behavior in the vicinity of the first-order and second-order regimes of the ferromagnet-paramagnet phase boundary, respectively. To achieve high-precision results, we utilize a combination of (i) a parallel version of the multicanonical algorithm and (ii) a hybrid updating scheme combining Metropolis and generalized Wolff cluster moves. These techniques are combined to study for the first time the correlation length of the model, using its scaling in the regime of second-order transitions to illustrate universality through the observed identity of the limiting value of ξ/L with the exactly known result for the Ising universality class.

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References

  1. M. Blume, Phys. Rev. 141, 517 (1966)

    Article  ADS  Google Scholar 

  2. H.W. Capel, Physica (Utr.) 32, 966 (1966)

    Article  ADS  Google Scholar 

  3. H.W. Capel, Physica (Utr.) 33, 295 (1967)

    Article  ADS  Google Scholar 

  4. H.W. Capel, Physica (Utr.) 37, 423 (1967)

    Article  ADS  Google Scholar 

  5. I.D. Lawrie, S. Sarbach, in Phase Transitions and Critical Phenomena, edited by C. Domb, J.L. Lebowitz., Vol. 9 (Academic Press, London, 1984)

  6. W. Selke, J. Oitmaa, J. Phys. C 22, 076004 (2010)

    Google Scholar 

  7. N.G. Fytas, Eur. Phys. J. B 79, 21 (2011)

    Article  ADS  Google Scholar 

  8. J. Zierenberg, N.G. Fytas, W. Janke, Phys. Rev. E 91, 032126 (2015)

    Article  ADS  Google Scholar 

  9. A.N. Berker, M. Wortis, Phys. Rev. B 14, 4946 (1976)

    Article  ADS  Google Scholar 

  10. D.P. Landau, Phys. Rev. Lett. 28, 449 (1972)

    Article  ADS  Google Scholar 

  11. M. Kaufman, R.B. Griffiths, J.M. Yeomans, M. Fisher, Phys. Rev. B 23, 3448 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  12. W. Selke, J. Yeomans, J. Phys. A 16, 2789 (1983)

    Article  ADS  Google Scholar 

  13. W. Selke, D.A. Huse, D.M. Kroll, J. Phys. A 17, 3019 (1984)

    Article  ADS  Google Scholar 

  14. D.P. Landau, R.H. Swendsen, Phys. Rev. B 33, 7700 (1986)

    Article  ADS  Google Scholar 

  15. J.C. Xavier, F.C. Alcaraz, D. Pena Lara, J.A. Plascak, Phys. Rev. B 57, 11575 (1998)

    Article  ADS  Google Scholar 

  16. Y. Deng, W. Guo, H.W.J. Blöte, Phys. Rev. E 72, 016101 (2005)

    Article  ADS  Google Scholar 

  17. C.J. Silva, A.A. Caparica, J.A. Plascak, Phys. Rev. E 73, 036702 (2006)

    Article  ADS  Google Scholar 

  18. D. Hurt, M. Eitzel, R.T. Scalettar, G.G. Batrouni, in Computer Simulation Studies in Condensed Matter Physics XVII, Springer Proceedings in Physics, edited by D.P. Landau, S.P. Lewis, H.-B. Schüttler, Vol. 105 (Springer, Berlin, 2007)

  19. A. Malakis, A.N. Berker, I.A. Hadjiagapiou, N.G. Fytas, Phys. Rev. E 79, 011125 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  20. A. Malakis, A.N. Berker, I.A. Hadjiagapiou, N.G. Fytas, T. Papakonstantinou, Phys. Rev. E 81, 041113 (2010)

    Article  ADS  Google Scholar 

  21. W. Kwak, J. Jeong, J. Lee, D.-H. Kim, Phys. Rev. E 92, 022134 (2015)

    Article  ADS  Google Scholar 

  22. M.J. Stephen, J.L. McColey, Phys. Rev. Lett. 44, 89 (1973)

    Article  Google Scholar 

  23. T.S. Chang, G.F. Tuthill, H.E. Stanley, Phys. Rev. B 9, 4482 (1974)

    ADS  Google Scholar 

  24. G.F. Tuthill, J.F. Nicoll, H.E. Stanley, Phys. Rev. B 11, 4579 (1975)

    Article  ADS  Google Scholar 

  25. F.J. Wegner, Phys. Lett. 54A, 1 (1975)

    Article  ADS  Google Scholar 

  26. P.F. Fox, A.J. Guttmann, J. Phys. C 6, 913 (1973)

    Article  ADS  Google Scholar 

  27. W.J. Camp, J.P. Van Dyke, Phys. Rev. B 11, 2579 (1975)

    Article  ADS  Google Scholar 

  28. T.W. Burkhardt, R.H. Swendsen, Phys. Rev. B 13, 3071 (1976)

    Article  ADS  Google Scholar 

  29. P.D. Beale, Phys. Rev. B 33, 1717 (1986)

    Article  ADS  Google Scholar 

  30. T.W. Burkhardt, Phys. Rev. B 14, 1196 (1976)

    Article  ADS  Google Scholar 

  31. T.W. Burkhardt, H.J.F. Knops, Phys. Rev. B. 15, 1602 (1977)

    Article  ADS  Google Scholar 

  32. M. Kaufman, R.B. Griffiths, J.M. Yeomans, M.E. Fisher, Phys. Rev. B 23, 3448 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  33. J.M. Yeomans, M.E. Fisher, Phys. Rev. B 24, 2825 (1981)

    Article  ADS  Google Scholar 

  34. N.B. Wilding, P. Nielaba, Phys. Rev. E 53, 926 (1996)

    Article  ADS  Google Scholar 

  35. J.A. Plascak, P.H.L. Martins, Comput. Phys. Commun. 184, 259 (2013)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  38. W. Janke, First-order phase transitions, in Computer Simulations of Surfaces and Interfaces, NATO Science Series, II. Mathematics, Physics and Chemistry – Vol. 114, Proceedings of the NATO Advanced Study Institute, Albena, Bulgaria, 9–20 September 2002, B. Dünweg, D.P. Landau, [ A.I.Milchev, eds. (Kluwer, Dordrecht, 2003), pp. 111–135

  39. W. Janke, R. Villanova, Nucl. Phys. B 489, 679 (1997)

    Article  ADS  Google Scholar 

  40. D.P. Landau, K. Binder, Monte Carlo Simulations in Statistical Physics (Cambridge University Press, Cambridge, 2000)

  41. J. Cardy, Scaling and Renormalization in Statistical Physics (Cambridge University Press, Cambridge, 1996)

  42. A. Pelissetto, E. Vicari, Phys. Rep. 368, 549 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  43. J. Salas, A.D. Sokal, J. Stat. Phys. 98, 551 (2000)

    Article  Google Scholar 

  44. W. Selke, L.N. Shchur, J. Phys. A 38, L739 (2005)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  47. W. Janke, Int. J. Mod. Phys. C 03, 1137 (1992)

    Article  ADS  Google Scholar 

  48. W. Janke, Physica A 254, 164 (1998)

    Article  ADS  Google Scholar 

  49. J. Zierenberg, M. Marenz, W. Janke, Comput. Phys. Comm. 184, 1155 (2013)

    Article  ADS  Google Scholar 

  50. J. Zierenberg, M. Marenz, W. Janke, Physics Procedia 53, 55 (2014)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  53. H.W.J. Blöte, E. Luijten, J.R. Heringa, J. Phys. A: Math. Gen. 28, 6289 (1995)

    Article  ADS  Google Scholar 

  54. M. Hasenbusch, Phys. Rev. B 82, 174433 (2010)

    Article  ADS  Google Scholar 

  55. A. Malakis, A.N. Berker, N.G. Fytas, T. Papakonstantinou, Phys. Rev. E 85, 061106 (2012)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  57. B. Efron, The Jackknife, the Bootstrap and other Resampling Plans (Society for Industrial and Applied Mathematics, Philadelphia, 1982)

  58. B. Efron, R.J. Tibshirani, An Introduction to the Bootstrap (Chapman and Hall, Boca Raton, 1994)

  59. F. Cooper, B. Freedman, D. Preston, Nucl. Phys. B 210, 210 (1982)

    Article  ADS  Google Scholar 

  60. H.G. Ballesteros, L.A. Fernández, V. Martín-Mayor, A. Muñoz Sudupe, G. Parisi, J.J. Ruiz-Lorenzo, J. Phys. A: Math. Gen. 32, 1 (1999)

    Article  ADS  Google Scholar 

  61. P. Young, Everything You Wanted to Know About Data Analysis and Fitting but Were Afraid to Ask, SpringerBriefs in Physics (Springer, Berlin, 2015)

  62. D. Amit, V. Martín-Mayor, Field Theory, the Renormalization Group and Critical Phenomena, 3rd edn. (World Scientific, Singapore, 2005)

  63. M.P. Nightingale, Physica (Amsterdam) 83A, 561 (1976)

    Article  ADS  Google Scholar 

  64. H.G. Ballesteros, L.A. Fernández, V. Martín-Mayor, A. Muñoz-Sudupe, Phys. Lett. B 378, 207 (1996)

    Article  ADS  Google Scholar 

  65. N.G. Fytas, V. Martín-Mayor, Phys. Rev. Lett. 110, 227201 (2013)

    Article  ADS  Google Scholar 

  66. P.E. Theodorakis, N.G. Fytas, Phys. Rev. E 86, 011140 (2012)

    Article  ADS  Google Scholar 

Download references

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Correspondence to Nikolaos G. Fytas.

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Zierenberg, J., Fytas, N.G., Weigel, M. et al. Scaling and universality in the phase diagram of the 2D Blume-Capel model. Eur. Phys. J. Spec. Top. 226, 789–804 (2017). https://doi.org/10.1140/epjst/e2016-60337-x

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