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Locally self-similar phase diagram of the disordered Potts model on the hierarchical lattice

J-Ch. Anglès d’Auriac1 and Ferenc Iglói2,3,*

  • 1Institut Néel-MCBT CNRS U.P.R. 5001 du CNRS, Laboratoire conventionné avec l’Université Joseph Fourier, B. P. 166, F-38042 Grenoble, France
  • 2Wigner Research Centre, Institute for Solid State Physics and Optics, H-1525 Budapest, P.O. Box 49, Hungary
  • 3Institute of Theoretical Physics, Szeged University, H-6720 Szeged, Hungary

  • *igloi.ferenc@wigner.mta.hu
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Phys. Rev. E 87, 022103 – Published 4 February, 2013

DOI: https://doi.org/10.1103/PhysRevE.87.022103

Abstract

We study the critical behavior of the random q-state Potts model in the large-q limit on the diamond hierarchical lattice with an effective dimensionality deff>2. By varying the temperature and the strength of the frustration the system has a phase transition line between the paramagnetic and the ferromagnetic phases which is controlled by four different fixed points. According to our renormalization group study the phase boundary in the vicinity of the multicritical point is self-similar; it is well represented by a logarithmic spiral. We expect an infinite number of reentrances in the thermodynamic limit; consequently one cannot define standard thermodynamic phases in this region.

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References (28)

  1. For reviews on spin glasses, see K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801 (1986); M. Mezard, G. Parisi, and M. A. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987); K. H. Fisher and J. A. Hertz, Spin Glasses (Cambridge University Press, Cambridge, 1991); Spin Glasses and Random Fields, edited by A. P. Young (World Scientific, Singapore, 1998).
  2. D. Elderfield and D. Sherrington, J. Phys. C 16, L497 (1983).
  3. D. J. Gross, I. Kanter, and H. Sompolinsky, Phys. Rev. Lett. 55, 304 (1985).
  4. H.-O. Carmesin and K. Binder, J. Phys. A 21, 4053 (1988).
  5. M. Scheucher, J. D. Reger, K. Binder, and A. P. Young, Phys. Rev. B 42, 6881 (1990).
  6. G. Schreider and J. D. Reger, J. Phys. A 28, 317 (1995).
  7. O. Dillmann, W. Janke, and K. Binder, J. Stat. Phys. 92, 57 (1998).
  8. S. F. Edwards and P. W. Anderson, J. Phys. F 5, 965 (1975).
  9. K. Binder and J. D. Reger, Adv. Phys. 41, 547 (1992).
  10. G. Parisi, in Complex Behavior of Glassy Systems, edited by M. Rubi and C. Perez-Vicente (Springer, Berlin, 1997), p. 111; S. Franz and G. Parisi, Physica A 261, 317 (1998); M. Mézard and G. Parisi, Phys. Rev. Lett. 82, 747 (1999); M. Mézard, Physica A 265, 352 (1999); G. Parisi, ibid. 280, 115 (2000); X. Xia and P. G. Wolynes, Phys. Rev. Lett. 86, 5526 (2001).
  11. C. Brangian, W. Kob, and K. Binder, J. Phys. A 36, 10847 (2003).
  12. L. W. Lee, H. G. Katzgraber, and A. P. Young, Phys. Rev. B 74, 104416 (2006).
  13. R. A. Banos et al., J. Stat. Mech. (2010) P05002.
  14. R. S. Andrist, D. Larson, and H. G. Katzgraber, Phys. Rev. E 83, 030106(R) (2011).
  15. A. A. Migdal, Sov. Phys. JETP 42, 743 (1976); L. P. Kadanoff, Ann. Phys. (NY) 100, 359 (1976).
  16. A. N. Berker and S. Ostlund, J. Phys. C 12, 4961 (1979); M. Kaufman and R. B. Griffiths, Phys. Rev. B 24, 496 (1981); R. B. Griffiths and M. Kaufman, ibid. 26, 5022 (1982).
  17. W. Kinzel and E. Domany, Phys. Rev. B 23, 3421 (1981).
  18. B. Derrida and E. Gardner, J. Phys. A 17, 3223 (1984); B. Derrida, in Critical Phenomena, Random Systems, Gauge Theories, edited by K. Osterwalder and R. Stora, Proceedings of the Les Houches Summer School of Theoretical Physics, 1984 (North- Holland, Amsterdam, 1986), p. 989.
  19. D. Andelman and A. N. Berker, Phys. Rev. B 29, 2630 (1984).
  20. C. Monthus and T. Garel, Phys. Rev. B 77, 134416 (2008).
  21. A. Benyoussef and M. Loulidi, Phys. Rev. B 53, 8215 (1996).
  22. F. Iglói and L. Turban, Phys. Rev. B 80, 134201 (2009).
  23. J.-Ch. Anglès d’Auriac and F. Iglói, Phys. Rev. Lett. 90, 190601 (2003); M.-T. Mercaldo, J.-Ch. Anglès d’Auriac, and F. Iglói, Phys. Rev. E 73, 026126 (2006).
  24. E. S. Sorensen, M. J. P. Gingras, and D. A. Huse, Europhys. Lett. 44, 504 (1998).
  25. I. Morgenstern and K. Binder, Phys. Rev. B 22, 288 (1980); I. Morgenstern and H. Horner, ibid. 25, 504 (1982); W. L. McMillan, ibid. 28, 5216 (1983).
  26. J. L. Jacobsen and M. Picco, Phys. Rev. E 65, 026113 (2002).
  27. For a review see W. Selke, Phys. Rep. 170, 213 (1988).
  28. B. Derrida, L. De Seze, and C. Itzykson, J. Stat. Phys. 33, 559 (1983).

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Phys. Rev. E 87, 022103– Published 4 February, 2013
  • Received 26 November 2012

©2013 American Physical Society

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