- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Restoration of dimensional reduction in the random-field Ising model at five dimensions

Nikolaos G. Fytas1, Víctor Martín-Mayor2,3, Marco Picco4, and Nicolas Sourlas5

  • 1Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, United Kingdom
  • 2Departamento de Física Téorica I, Universidad Complutense, 28040 Madrid, Spain
  • 3Instituto de Biocomputacíon y Física de Sistemas Complejos (BIFI), 50009 Zaragoza, Spain
  • 4Sorbonne Universités, Université Pierre et Marie Curie–Paris VI, Laboratoire de Physique Théorique et Hautes Energies, 4, Place Jussieu, 75252 Paris Cedex 05, France
  • 5Laboratoire de Physique Théorique de l'Ecole Normale Supérieure (Unité Mixte de Recherche du CNRS et de l'Ecole Normale Supérieure, associée à l'Université Pierre et Marie Curie, PARIS VI) 24 rue Lhomond, 75231 Paris Cedex 05, France

PDF

Phys. Rev. E 95, 042117 – Published 10 April, 2017

DOI: https://doi.org/10.1103/PhysRevE.95.042117

Abstract

The random-field Ising model is one of the few disordered systems where the perturbative renormalization group can be carried out to all orders of perturbation theory. This analysis predicts dimensional reduction, i.e., that the critical properties of the random-field Ising model in D dimensions are identical to those of the pure Ising ferromagnet in D−2 dimensions. It is well known that dimensional reduction is not true in three dimensions, thus invalidating the perturbative renormalization group prediction. Here, we report high-precision numerical simulations of the 5D random-field Ising model at zero temperature. We illustrate universality by comparing different probability distributions for the random fields. We compute all the relevant critical exponents (including the critical slowing down exponent for the ground-state finding algorithm), as well as several other renormalization-group invariants. The estimated values of the critical exponents of the 5D random-field Ising model are statistically compatible to those of the pure 3D Ising ferromagnet. These results support the restoration of dimensional reduction at D=5. We thus conclude that the failure of the perturbative renormalization group is a low-dimensional phenomenon. We close our contribution by comparing universal quantities for the random-field problem at dimensions 3≤D<6 to their values in the pure Ising model at D−2 dimensions, and we provide a clear verification of the Rushbrooke equality at all studied dimensions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (40)

  1. G. Parisi, Field Theory, Disorder and Simulations (World Scientific, Singapore, 1994).
  2. S. F. Edwards and P. W. Anderson, J. Phys. F 5, 965 (1975).
  3. A. P. Young, J. Phys. C 10, L257 (1977).
  4. G. Parisi and N. Sourlas, Phys. Rev. Lett. 43, 744 (1979).
  5. J. Bricmont and A. Kupiainen, Phys. Rev. Lett. 59, 1829 (1987).
  6. N. G. Fytas, V. Martín-Mayor, M. Picco, and N. Sourlas, Phys. Rev. Lett. 116, 227201 (2016).
  7. M. Talagrand, Ann. Math. 163, 221 (2006).
  8. D. Panchenko, Ann. Math. 177, 383 (2013).
  9. G. Parisi and N. Sourlas, Phys. Rev. Lett. 89, 257204 (2002).
  10. E. Brézin and C. De Dominicis, Europhys. Lett. 44, 13 (1998).
  11. E. Brézin and C. De Dominicis, Eur. Phys. J. B 19, 467 (2001).
  12. M. Kardar and Y.-C. Zhang, Phys. Rev. Lett. 58, 2087 (1987).
  13. M. Kardar, Nucl. Phys. B 290, 582 (1987).
  14. E. Medina, M. Kardar, Y. Shapir, and X. R. Wang, Phys. Rev. Lett. 62, 941 (1989).
  15. E. Medina and M. Kardar, J. Stat. Phys. 71, 967 (1993).
  16. G. Parisi, M. Picco, and N. Sourlas, Europhys. Lett. 66, 465 (2004).
  17. P. Dimock and J.-P. Eckmann, Commun. Math. Phys. 51, 41 (1976).
  18. A. Aharony, Phys. Rev. B 18, 3318 (1978).
  19. M. Tissier and G. Tarjus, Phys. Rev. Lett. 107, 041601 (2011).
  20. M. Tissier and G. Tarjus, Phys. Rev. B 85, 104203 (2012).
  21. G. Tarjus, I. Balog, and M. Tissier, Europhys. Lett. 103, 61001 (2013).
  22. N. G. Fytas and V. Martín-Mayor, Phys. Rev. Lett. 110, 227201 (2013).
  23. N. G. Fytas and V. Martín-Mayor, Phys. Rev. E 93, 063308 (2016).
  24. J. Villain, Phys. Rev. Lett. 52, 1543 (1984).
  25. A. J. Bray and M. A. Moore, Phys. Rev. B 31, 631 (1985).
  26. D. S. Fisher and D. A. Huse, Phys. Rev. Lett. 56, 1601 (1986).
  27. D. J. Amit and V. Martín-Mayor, Field Theory, the Renormalization Group and Critical Phenomena, 3rd ed. (World Scientific, Singapore, 2005).
  28. B. Ahrens and A. K. Hartmann, Phys. Rev. B 83, 014205 (2011).
  29. A. V. Goldberg and R. E. Tarjan, J. ACM 35, 921 (1988).
  30. A. A. Middleton, Phys. Rev. Lett. 88, 017202 (2001).
  31. A. A. Middleton and D. S. Fisher, Phys. Rev. B 65, 134411 (2002).
  32. A. A. Middleton, arXiv:cond-mat/0208182.
  33. M. Nightingale, Physica A 83, 561 (1976).
  34. H. G. Ballesteros, L. A. Fernandez, V. Martín-Mayor, and A. Muñoz Sudupe, Phys. Lett. B 378, 207 (1996).
  35. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University Press, Cambridge, 1992).
  36. H. G. Ballesteros, L. A. Fernández, V. Martín-Mayor, A. Muñoz Sudupe, G. Parisi, and J. J. Ruiz-Lorenzo, Phys. Rev. B 58, 2740 (1998).
  37. F. Kos, D. Poland, D. Simmons-Duffin, and A. Vichi, J. High Energy Phys. 08 (2016) 036.
  38. A. T. Ogielski, Phys. Rev. B 32, 7384 (1985).
  39. N. G. Fytas, V. Martín-Mayor, M. Picco, and N. Sourlas, Specific-heat exponent and modified hyperscaling in the 4d random-field ising model, J. Stat. Mech. (2017) 033302.
  40. B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model (Harvard University Press, Cambridge, MA, 1973).

Outline

Information

Phys. Rev. E 95, 042117– Published 10 April, 2017
  • Received 19 December 2016

©2017 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation