Skip to main content
SpringerLink
Log in

Weak First-Order Transition and Pseudoscaling Behavior in the Universality Class of the O(N) Ising Model

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

Using Monte Carlo and renormalization group methods, we investigate systems with critical behavior described by two order parameters: continuous (vector) and discrete (scalar). We consider two models of classical three-dimensional Heisenberg magnets with different numbers of spin components N = 1,…,4: the model on a cubic lattice with an additional competing antiferromagnetic exchange interaction in a layer and the model on a body-centered lattice with two competing antiferromagnetic interactions. In both models, we observe a first-order transition for all values of N. In the case where competing exchanges are approximately equal, the first order of a transition is close to the second order, and pseudoscaling behavior is observed with critical exponents differing from those of the O(N) model. In the case N = 2, the critical exponents are consistent with the well-known indices of the class of magnets with a noncollinear spin ordering. We also give a possible explanation of the observed pseudoscaling in the framework of the renormalization group analysis.

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. D. Loison, “Phase transitions in frustrated vector spin systems: Numerical studies,” in: Frustrated Spin Systems (H. T. Diep, ed.), World Scientific, Singapore (2004), pp. 177–228.

    Google Scholar 

  2. B. Delamotte, D. Mouhanna, and M. Tissier, “Nonperturbative renormalization-group approach to frustrated magnets,” Phys. Rev. B, 69, 134413 (2004); arXiv:cond-mat/0309101v1 (2003).

    Article  ADS  MATH  Google Scholar 

  3. O. A. Starykh, “Unusual ordered phases of highly frustrated magnets: A review,” Rep. Prog. Phys., 78, 052502 (2015); arXiv:1412.8482v1 [cond-mat.str-el] (2014).

    Article  ADS  Google Scholar 

  4. H. Kawamura, “Phase transitions in Heisenberg antiferromagnets on triangular and layered-triangular lattices (invited),” J. Appl. Phys., 61, 3590–3594 (1987); “Critical properties of helical magnets and triangular antifer-romagnets,” J. Appl. Phys., 63, 3086–3088 (1988).

    Article  ADS  Google Scholar 

  5. G. Zumbach, “Almost second order phase transitions,” Phys. Rev. Lett., 71, 2421–2424 (1993).

    Article  ADS  Google Scholar 

  6. M. Tiesser, B. Delamotte, and D. Mouhanna, “Frustrated heisenberg magnets: A nonperturbative approach,” Phys. Rev. Lett., 84, 5208–5211 (2000); arXiv:cond-mat/0001350v1 (2000).

    Article  ADS  Google Scholar 

  7. M. Plischke and J. Oitmaa, “Ising models with n 1: A series-expansion approach,” Phys. Rev. B, 19, 487–493 (1979).

    Article  ADS  Google Scholar 

  8. J. R. Banavar, D. Jasnow, and D. P. Landau, “Fluctuation-induced first-order transition in a bcc Ising model with competing interactions,” Phys. Rev. B, 20, 3820–3827 (1979).

    Article  ADS  Google Scholar 

  9. M. J. Velgakis and M. Ferer, “Fluctuation-induced, first-order transition in a bcc Ising model with competing interactions,” Phys. Rev. B, 27, 401–412 (1983).

    Article  ADS  Google Scholar 

  10. P. Butera and M. Comi, “Critical universality and hyperscaling revisited for Ising models of general spin using extended high-temperature series,” Phys. Rev. B, 65, 144431 (2002); arXiv:hep-lat/0112049v1 (2001).

    Article  ADS  Google Scholar 

  11. Y. Kamiya, N. Kawashima, and C. D. Batista, “Dimensional crossover in the quasi-two-dimensional Ising-O(3) model,” Phys. Rev. B, 84, 214429 (2011); arXiv:1108.1599v3 [cond-mat.str-el] (2011).

    Article  ADS  Google Scholar 

  12. A. O. Sorokin and A. V. Syromyatnikov, “Ising–Heisenberg universality class in three-dimensional frustrated magnetic systems [in Russian],” Communication No. 2939, Petersburg Inst. Nucl. Phys., Gatchina (2013).

    Google Scholar 

  13. A. O. Sorokin, “Critical and multicritical behavior in the Ising–Heisenberg universality class,” Phys. Lett. A, 382, 3455–3462 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  14. A. O. Sorokin, “Ising–XY transition in three-dimensional frustrated antiferromagnets with collinear spin ordering,” JETP Lett., 109, 419–423 (2019).

    Article  ADS  Google Scholar 

  15. M. K. Ramazanov and A. K. Murtazaev, “Phase transitions and critical characteristics in the layered antiferro-magnetic Ising model with next-nearest-neighbor intralayer interactions,” JETP Lett., 101, 714–718 (2015).

    Article  ADS  Google Scholar 

  16. A. K. Murtazaev, M. K. Ramazanov, F. A. Kassan-Ogly, and D. R. Kurbanova, “Phase transitions in the antifer-romagnetic Ising model on a body-centered cubic lattice with interactions between next-to-nearest neighbors,” JETP, 120, 110–114 (2015).

    Article  ADS  Google Scholar 

  17. A. K. Murtazaev, M. K. Ramazanov, D. R. Kurbanova, M. K. Badiev, and Ya. K. Abuev, “A study of the critical properties of the Ising model on body-centered cubic lattice taking into account the interaction of next behind nearest neighbors,” Phys. Solid State, 59, 1103–1109 (2017).

    Article  ADS  Google Scholar 

  18. M. K. Ramazanov and A. K. Murtazaev, “Phase transitions and critical properties in the antiferromagnetic Heisenberg model on a layered cubic lattice,” JETP Lett., 106, 86–91 (2017).

    Article  ADS  Google Scholar 

  19. A. O. Sorokin and A. V. Syromyatnikov, “Transitions in three-dimensional magnets with extra broken symmetry,” Solid State Phenomena, 190, 63–66 (2012).

    Article  Google Scholar 

  20. C. L. Henley, “Ordering due to disorder in a frustrated vector antiferromagnet,” Phys. Rev. Lett., 62, 2056–2059 (1989).

    Article  ADS  Google Scholar 

  21. E. F. Shender, “Antiferromagnetic garnets with fluctuationally interacting sublattices,” Sov. Phys. JETP, 56, 178–184 (1982).

    Google Scholar 

  22. E. Brezin, J. C. Le Guillou, and J. Zinn-Justin, “Discussion of critical phenomena for general n-vector models,” Phys. Rev. B, 10, 892–900 (1974).

    Article  ADS  Google Scholar 

  23. K. De’Bell and D. J. W. Geldart, “Coefficients to O(ε 3) for the mixed fixed point of the nm-component field model,” Phys. Rev. B, 32, 4763–4765 (1985).

    Article  ADS  Google Scholar 

  24. Y. M. Pis’mak, A. Weber, and F. J. Wegner, “Critical behavior of a general O(n)-symmetric model of two n-vector fields in D = 4 − 2,” J. Phys. A: Math. Theor., 42, 095003 (2009); arXiv:0809.1568v2 [cond-mat.stat-mech] (2008).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. P. Bak and D. Mukamel, “Physical realizations of n 4-component vector models: III. Phase transitions in Cr, Eu, MnS2, Ho, Dy, and Tb,” Phys. Rev. B, 13, 5086–5094 (1976).

    Article  ADS  Google Scholar 

  26. S. A. Brazovskii, I. E. Dzyaloshinskii, and B. G. Kukharenko, “First-order magnetic phase transitions and fluctuations,” Sov. Phys. JETP, 43, 1178–1183 (1976).

    ADS  Google Scholar 

  27. I. E. Dzyaloshinskii, “Character of phase transitions to a helical or sinusoidal state in magnetic materials,” Sov. Phys. JETP, 45, 1014–1022 (1977).

    ADS  Google Scholar 

  28. D. R. Nelson, J. M. Kosterlitz, and M. E. Fisher, “Renormalization-group analysis of bicritical and tetracritical points,” Phys. Rev. Lett., 33, 813–817 (1974).

    Article  ADS  Google Scholar 

  29. I. F. Lyuksyutov, V. L. Pokrovskii, and D. E. Khmel’nitskii, “Intersection of lines of second-order transitions,” Sov. Phys. JETP, 42, 923–926 (1976).

    ADS  Google Scholar 

  30. J. M. Kosterlitz, D. R. Nelson, and M. E. Fisher, “Bicritical and tetracritical points in anisotropic antiferromagnetic systems,” Phys. Rev. B, 13, 412–432 (1976).

    Article  ADS  Google Scholar 

  31. T. Garel and P. Pfeuty, “Commensurability effects on the critical behaviour of systems with helical ordering,” J. Phys. C, 9, L245–L249 (1976).

    Article  ADS  Google Scholar 

  32. Z. Barak and M. B. Walker, “First-order phase transitions in Tb, Dy, and Ho,” Phys. Rev. B, 25, 1969–1972 (1982).

    Article  ADS  Google Scholar 

  33. H. Kawamura, “Renormalization-group analysis of chiral transitions,” Phys. Rev. B, 38, 4916–4928 (1988).

    Article  ADS  Google Scholar 

  34. P. Calabrese, A. Pelissetto, and E. Vicari, “Multicritical phenomena in O(n 1) ⊕ O(n 2)-symmetric theories,” Phys. Rev. B, 67, 054505 (2003); arXiv:cond-mat/0209580v2 [cond-mat.stat-mech] (2002).

    Article  ADS  Google Scholar 

  35. S. A. Antonenko, A. I. Sokolov, and K. B. Varnashev, “Chiral transitions in three-dimensional magnets and higher order ε expansion,” Phys. Lett. A, 208, 161–164 (1995); arXiv:cond-mat/9803377v1 [cond-mat.stat-mech] (1998).

    Article  ADS  Google Scholar 

  36. A. Pelissetto, P. Rossi, and E. Vicari, “Large-n critical behavior of O(n) × O(m) spin models,” Nucl. Phys. B, 607, 605–634 (2001); arXiv:hep-th/0104024v2 (2001).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. P. Calabrese and P. Parruccini, “Five-loop ε expansion for O(n) × O(m) spin models,” Nucl. Phys. B, 679, 568–596 (2004); arXiv:cond-mat/0308037v3 (2003).

    Article  ADS  MATH  Google Scholar 

  38. F. R. Brown and T. J. Woch, “Overrelaxed heat-bath and Metropolis algorithms for accelerating pure gauge Monte Carlo calculations,” Phys. Rev. Lett., 58, 2394–2396 (1987).

    Article  ADS  Google Scholar 

  39. M. Creutz, “Overrelaxation and Monte Carlo simulation,” Phys. Rev. D, 36, 515–519 (1987).

    Article  ADS  MathSciNet  Google Scholar 

  40. K. Binder, “Finite size scaling analysis of Ising model block distribution functions,” Z. Phys. B, 43, 119–140 (1981); “Critical properties from Monte Carlo coarse graining and renormalization,” Phys. Rev. Lett., 47, 693–696 (1981).

    Article  ADS  Google Scholar 

  41. A. M. Ferrenberg and D. P. Landau, “Critical behavior of the three-dimensional Ising model: A high-resolution Monte Carlo study,” Phys. Rev. B, 44, 5081–5091 (1991).

    Article  ADS  Google Scholar 

  42. H. Kawamura, “Monte Carlo study of chiral criticality–XY and Heisenberg stacked-triangular antiferromag-nets,” J. Phys. Soc. Japan, 61, 1299–1325 (1992).

    Article  ADS  Google Scholar 

  43. A. O. Sorokin, “Critical behavior of three-dimensional frustrated helimagnets,” JETP, 118, 417–425 (2014).

    Article  ADS  Google Scholar 

  44. A. O. Sorokin and A. V. Syromyatnikov, “First order transition in three-dimensional systems with fully broken O(3) symmetry,” JETP, 112, 1004–1012 (2011).

    Article  ADS  Google Scholar 

  45. A. O. Sorokin and A. V. Syromyatnikov, “Transitions in three-dimensional XY magnets with two chiral order parameters,” JETP, 113, 673–677 (2011).

    Article  ADS  Google Scholar 

  46. M. Tiesser, B. Delamotte, and D. Mouhanna, “XY frustrated systems: Continuous exponents in discontinuous phase transitions,” Phys. Rev. B, 67, 134422 (2003); arXiv:cond-mat/0107183v2 (2001).

    Article  ADS  Google Scholar 

  47. M. L. Plumer and A. Mailhot, “Tricritical behavior of the frustrated XY antiferromagnet,” Phys. Rev. B, 50, 16113–16116 (1994); arXiv:cond-mat/9405009v1 (1994).

    Article  ADS  Google Scholar 

  48. S. Fujimoto, “Low-energy properties of two-dimensional quantum triangular antiferromagnets: Nonperturbative renormalization group approach,” Phys. Rev. B, 73, 184401 (2006); arXiv:cond-mat/0511215v2 (2005).

    Article  ADS  Google Scholar 

  49. N. Tetradis and C. Wetterich, “Critical exponents from the effective average action,” Nucl. Phys. B, 422, 541–592 (1994); arXiv:hep-ph/9308214v1 (1993).

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Petersburg Nuclear Physics Institute, National Research Center Kurchatov Institute, Gatchina, Leningrad Oblast, Russia

    A. O. Sorokin

Authors
  1. A. O. Sorokin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. O. Sorokin.

Additional information

Conflicts of Interest. The author declares no conflicts of interest.

This research is supported by the Russian Foundation for Basic Research (Grant No. 16-32-60143).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sorokin, A.O. Weak First-Order Transition and Pseudoscaling Behavior in the Universality Class of the O(N) Ising Model. Theor Math Phys 200, 1193–1204 (2019). https://doi.org/10.1134/S0040577919080117

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040577919080117

Keywords

Search

Navigation