Journal of Physical Studies 28(2), Article 2601 [12 pages] (2024)
DOI: https://doi.org/10.30970/jps.28.2601

CRITICAL BEHAVIOR OF STRUCTURALLY DISORDERED SYSTEMS WITH LONG-RANGE INTERACTION

M. Dudka1,2,3 , D. Shapoval1,2 , Yu. Holovatch1,2,4,5 

1 Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, Lviv, UA–79011 Ukraine,
2 𝕃4 Collaboration & Doctoral College for the Statistical Physics of Complex Systems, Leipzig–Lorraine–Lviv–Coventry, Europe,
3 Lviv Polytechnic National University, Lviv, UA–79013, Ukraine,
4 Centre for Fluid and Complex Systems, Coventry University, Coventry CV15FB, UK,
5 Complexity Science Hub Vienna, 1080 Vienna, Austria

Received 29 November 2023; in final form 19 February 2024; accepted 28 February 2024; published online 07 May 2024

Some physical systems, although different by their microscopic nature, in the vicinity of critical points can have similar thermodynamic and structural properties governed by universal power laws. Such systems are grouped into the so-called universality classes. The latter are determined by such global characteristics as the type of interaction, space dimensionality, symmetry, and the number of components of the order parameter. Therefore, it is interesting to investigate how the mutual influence of these characteristics can affect the critical features of the system. The goal of our research is to analyze the changes in the critical behavior of a many-particle magnetic system under the influence of two competing factors: long-range interaction and weak structural disorder. Using the example of an $n$-vector model in a $d$-dimensional space, we will investigate ferromagnetic ordering in a structurally disordered magnet with the long-range interaction decaying with the distance $x$ as $J(x) \sim x^{- d - σ}$, with the control parameter $σ$. The analysis is carried out using the field-theoretical renormalization group methods, which makes it possible to identify the universality classes of the system under consideration, their change with global parameters, and to determine the universal characteristics of critical behavior (critical exponents and marginal dimensions). It will be shown that there exists such a region of parameters $(d,n,σ)$, where the long-range interaction and structural disorder lead to a synergistic effect and the appearance of a new, “random long-range” universality class. We obtain the renormalization group functions in the three-loop approximation, and calculate the correlation length critical exponent $ν(ϵ',n)$ as a perturbation theory series in $ϵ' = 2σ - d$. Quantitative estimates are obtained using asymptotic series resummation methods.

Key words: critical properties, universality classes, quenched disorder, long-range interactions, renormalization group, resummation.

Full text

References
  1. C. Domb, The Critical Point (Taylor & Francis, London, 1996);
    Crossref
  2. Order, Disorder and Criticality. Advanced Problems of Phase Transition Theory. Vol.7, edited by Yu. Holovatch (World Scientific, Singapore, 2004–2022);
    Crossref
  3. Yu. Holovatch et al. J. Phys. Stud. 22, 2801 (2018);
    Crossref
  4. M. Henkel, H. Hinrichsen, S. Lübeck, Non-Equilibrium Phase Transitions (Heidelber, Springer, 2008);
    Crossref
  5. G. Odor, Universality in Nonequilibrium Lattice Systems: Theoretical Foundations (Singapore, World Scientific, 2008);
    Crossref
  6. R. Folk, G. Moser, J. Phys. A: Math. Gen. 39, R207 (2006);
    Crossref
  7. I. Herbut, A Modern Approach to Critical Phenomena (Cambridge University Press, 2007);
    Crossref
  8. M. A. Anisimov, A. A. Povodyrev, V. D. Kulikov, J. V. Sengers, Phys. Rev. Lett. 75, 3146 (1995);
    Crossref
  9. M. Anisimov, A. Povodyrev, J. Sengers, Fluid Phase Equilib. 158– 160, 537(1999);
    Crossref
  10. V. S. Dotsenko, Phys. Usp. 38, 457 (1995);
    Crossref
  11. Yu. Holovatch et al., Int. J. Mod. Phys. B 16, 4027 (2002);
    Crossref
  12. N. Defenu, A. Trombettoni, A. Codello, Phys. Rev. E 92, 052113 (2015);
    Crossref
  13. N. Defenu, A. Codello, S. Ruffo, A. Trombettoni, J. Phys. A 53 143001 (2020);
    Crossref
  14. D. J. Amit, Field Theory, the Renormalization Group, and Critical Phenomena (World Scientific, Singapore, 1989);
    Crossref
  15. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford University Press, Oxford, 1996);
    Crossref
  16. H. Kleinert, V. Schulte-Frohlinde, Critical Properties of ϕ4-Theories (World Scientific, Singapore, 2001);
    Crossref
  17. Dynamics and Thermodynamics of Systems with Long-Range Interactions, edited by T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens, Lect. Not. Phys. 602, (Springer-Verlag, New York, 2002);
    Crossref
  18. A. Campa, T. Dauxois. S. Ruo, Phys. Rep. 480, 57 (2009);
    Crossref
  19. A. Cavagna et. al., Phys. Rev. E 92 012705, (2015);
    Crossref
  20. T. Padmanabhan, Phys. Rep. 188, 285 (1990);
    Crossref
  21. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon Press, London, 1960).
  22. D. Nicholson, Introduction to Plasma Theory (Wiley, New-York, 1983);
    Crossref
  23. A. Campa, T. Dauxois, D. Fanelli, S. Ruffo, Physics of Long-Range Interacting Systems (Oxford University Press, Oxford, 2014);
    Crossref
  24. D. Mukamel, preprint arXiv: 0905.1457 (2009);
    Crossref
  25. E. Bayong, H. T. Diep, V. Dotsenko, Phys. Rev. Lett. 83, 14 (1999);
    Crossref
  26. N. Defenu et al., Rev. Mod. Phys. 95, 035002 (2023);
    Crossref
  27. M. E. Fisher, S. K. Ma, and B. G. Nickel, Phys. Rev. Lett. 29, 917 (1972);
    Crossref
  28. J. Sak, Phys. Rev. B 8, 281 (1973);
    Crossref
  29. F. Bouchet, S. Gupta, D. Mukamel, Physica A 389, 4389 (2010);
    Crossref
  30. A. Pelissetto, E. Vicari, Phys. Rept. 368, 549 (2002);
    Crossref
  31. M. Dudka, Yu. Holovatch, T. Yavorskii, Acta Phys. Slovaca 52, 323 (2002); http://www.physics.sk/aps/pubs/2002/aps-2002-52-4-323.pdf
  32. M. Dudka, Yu. Holovatch, T. Yavors'kii, J. Phys. A 37, 10727 (2004);
    Crossref
  33. Yu. Holovatch, M. Dudka, T. Yavors'kii, J. Phys. Stud. 5, 233 (2001);
    Crossref
  34. D. Ivaneyko, J. Ilnytskyi, B. Berche, Yu. Holovatch, Physica A 370, 163 (2006);
    Crossref
  35. D. Shapoval, M. Dudka, A. A. Fedorenko, Yu. Holovatch, Phys. Rev. B 101, 064402 (2020);
    Crossref
  36. T. Ising, R. Folk, R. Kenna, B. Berche, Yu. Holovatch, J. Phys. Stud. 21, 3002 (2017);
    Crossref
  37. H. E. Stanley, Phys. Rev. Lett., 20, 589 (1968);
    Crossref
  38. Yu. Holovatch, Condens. Matter Phys. 9, 237 (2006);
    Crossref
  39. K. G. Wilson, M. E. Fisher, Phys. Rev. Lett. 28, 240 (1972);
    Crossref
  40. B. Berche, T. Ellis, Yu. Holovatch, R. Kenna, SciPost Phys. Lect. Not. 60 (2022);
    Crossref
  41. M. V. Kompaniets, E. Panzer, Phys. Rev. D 96, 036016 (2017);
    Crossref
  42. O. Schnetz, Phys. Rev. D 97, 085018 (2018);
    Crossref
  43. O. Schnetz, Phys. Rev. D 107, 036002 (2023);
    Crossref
  44. J. C. Le Guillou, J. Zinn-Justin, Phys. Rev. B 21, 3976 (1980);
    Crossref
  45. B. Delamotte, Yu. Holovatch, D. Ivaneyko, D. Mouhanna, M. Tissier, J. Stat. Mech. P03014 (2008);
    Crossref
  46. B. Delamotte, M. Dudka, Yu. Holovatch, D. Mouhanna, Phys. Rev. B 82, 104432 (2010);
    Crossref
  47. B. Delamotte, M. Dudka, Yu. Holovatch, D. Mouhanna, Condens. Matter Phys. 13, 43703 (2010);
    Crossref
  48. G. S. Joyce, Phys.Rev. 146, 349 (1966);
    Crossref
  49. F. J. Dyson, Commun. Math. Phys. 12, 91 (1969);
    Crossref
  50. E. Luijten, H. W. J. Blöte, Phys. Rev. Lett. 89, 025703 (2002);
    Crossref
  51. E. Luijten, Ph.D. Thesis (Technische Universiteit Delft, Delft, The Netherlands, 1997); http://resolver.tudelft.nl/uuid:8ec8fc17-4a7e-47e3-aec0-bceedcfa5904
  52. C. Behan, J. Phys. A: Math. Theor. 52, 075401 (2019);
    Crossref
  53. D. Benedetti, R. Gurau, S. Harribey, K. Suzuki, J. Phys. A 53, 445008 (2020);
    Crossref
  54. R. Brout, Phys. Rev. 115, 824 (1959);
    Crossref
  55. A. B. Harris, J. Phys. C 7, 1671 (1974);
    Crossref
  56. V. J. Emery, Phys. Rev. B 11, 239 (1975);
    Crossref
  57. M. Mézard, G. Parisi, M. A. Virasoro, Spin Glass Theory and Beyond. An Introduction to the Replica Method and Its Applications (World Scientific, Singapore, 1987);
    Crossref
  58. G. Parisi, arXiv:cond-mat/9701068 (1997);
    Crossref
  59. V. Dotsenko, Introduction to the Replica Theory of Disordered Statistical Systems (Cambridge, Cambridge University Press, 2001);
    Crossref
  60. R. Folk, Yu. Holovatch, T. Yavors'kii, Phys.-Usp. 46, 169 (2003);
    Crossref
  61. D. E. Khmel'nitskii, Zh. Eksp. Teor. Fiz. 68, 1960 (1975) [Sov. Phys. JETP 41, 981 (1975)];
  62. T. C. Lubensky, Phys. Rev. B 11, 3573 (1975);
    Crossref
  63. G. Grinstein, A. Luther, Phys. Rev. B 13, 1329 (1976);
    Crossref
  64. R. Folk, Yu. Holovatch, T. Yavors'kii, Phys. Rev. B 61, 15114 (2000);
    Crossref
  65. G. Parisi, unpublished;
  66. G. Parisi, J. Stat. Phys. 23, 49 (1980);
    Crossref
  67. R. Schloms, V. Dohm, Europhys. Lett. 3, 413 (1987);
    Crossref
  68. R. Schloms, V. Dohm, Nucl. Phys. B 328, 639 (1989);
    Crossref
  69. Yu. Holovatch, M. Shpot, J. Stat. Phys. 66, 867 (1992);
    Crossref
  70. Yu. Holovatch, T. Yavors'kii, J. Stat. Phys. 92, 785 (1998);
    Crossref
  71. Y. Yamazaki, Physica A 90, 547 (1978);
    Crossref
  72. A. Theumann, J. Phys. A: Math. Gen. 14 2759 (1981);
    Crossref
  73. Yu. Holovatch, unpublished.
  74. D. Shapoval, M. Dudka, Yu. Holovatch, Low Temp. Phys. 48, 1049 (2022);
    Crossref
  75. D. Shapoval, M. Dudka, Yu. Holovatch, unpublished (2023).