Skip to main content
SpringerLink
Log in

The Fully Frustrated XY Model Revisited: A New Universality Class

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

The two-dimensional (2d) fully frustrated Planar Rotator model on a square lattice has been the subject of a long controversy due to the simultaneous \(Z_2\) and O(2) symmetry existing in the model. The O(2) symmetry being responsible for the Berezinskii–Kosterlitz–Thouless transition (BKT) while the \(Z_2\) drives an Ising-like transition. There are arguments supporting two possible scenarios, one advocating that the loss of Ising and BKT order take place at the same temperature \(T_{t}\) and the other that the \(Z_2\) transition occurs at a higher temperature than the BKT one. In the first case an immediate consequence is that this model is in a new universality class. Most of the studies take hand of some order parameter like the stiffness, Binder’s cumulant or magnetization to obtain the transition temperature. Considering that the transition temperatures are obtained, in general, as an average over the estimates taken about several of those quantities, it is difficult to decide if they are describing the same or slightly separate transitions. In this paper we describe an iterative method based on the knowledge of the complex zeros of the energy probability distribution to study the critical behavior of the system. The method is general with advantages over most conventional techniques since it does not need to identify any order parameter a priori. The critical temperature and exponents can be obtained with good precision. We apply the method to study the Fully Frustrated Planar Rotator (FFPR) and the Anisotropic Heisenberg (FFXY) models in two dimensions. We show that both models are in a new universality class with \(T_{PR}=0.45286(32)\) and \(T_{XY}=0.36916(16)\) respectively and the transition exponent \(\nu =0.824(30)\) (\(\frac{1}{\nu }=1.22(4)\)).

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Mermin, N.D., Wagner, H.: Absence of ferromagnetism or antiferromagnetism in one-or two-dimensional isotropic Heisenberg models. Phys. Rev. Lett. 17, 1133 (1966). https://doi.org/10.1103/PhysRevLett.17.1133

    Article  ADS  Google Scholar 

  2. Berezinskii, V.L.: Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group I. Classical systems. Sov. Phys. JETP 32, 493 (1971)

    MathSciNet  ADS  Google Scholar 

  3. Kosterlitz, J.M., Thouless, D.J.: Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C Solid State Phys. 6, 1181 (1973)

    Article  ADS  Google Scholar 

  4. José, J.V. (ed.): \(40\) Years History of Berezinskii-Kosterlitz-Thouless Theory. World Scientific, Singapore (2013)

    MATH  Google Scholar 

  5. Minnhagen, P.: The two-dimensional Coulomb gas, vortex unbinding, and superfluid-superconducting films. Rev. Mod. Phys. 59, 1001 (1987)

    Article  ADS  Google Scholar 

  6. Rocha, J.C.S., Mól, L.A.S., Costa, B.V.: Using zeros of the canonical partition function map to detect signatures of a Berezinskii–Kosterlitz–Thouless transition. Comput. Phys. Commun. (2016). https://doi.org/10.1016/j.cpc.2016.08.016

  7. Figueiredo, T.P., Rocha, J.C.S., Costa, B.V.: Topological phase transition in the two-dimensional anisotropic Heisenberg model: a study using the replica exchange Wang–Landau sampling. Physica A 488, 121 (2017). https://doi.org/10.1016/j.physa.2017.07.010

    Article  ADS  Google Scholar 

  8. Kenna, R., Irving, A.C.: The Kosterlitz-Thouless universality class. Nucl. Phys. 485, 583 (1997). https://doi.org/10.1016/S0550-3213(96)00642-6

    Article  ADS  Google Scholar 

  9. Villain, J.: Spin glass with non-random interactions. J. Phys. C 10, 1717 (1977)

    Article  ADS  Google Scholar 

  10. Villain, J.: Two-level systems in a spin-glass model. I. General formalism and two-dimensional model. J. Phys. C 10, 4793 (1977)

    Article  ADS  Google Scholar 

  11. Teitel, S., Jayaprakash, C.: Phase transtions in frustrated two-dimensional XY models. Phys. Rev. B 27, 598(R) (1983). https://doi.org/10.1103/PhysRevB.27.598

    Article  ADS  Google Scholar 

  12. Thijssen, J.M., Knops, H.J.F.: Monte Carlo transfer-matrix study of the frustrated XY model. Phys. Rev. B 42, 2438 (1990). https://doi.org/10.1103/PhysRevB.42.2438

    Article  ADS  Google Scholar 

  13. Granato, E., Nightingale, M.P.: Chiral exponents of the square-lattice frustrated XY model: a Monte Carlo transfer-matrix calculation. Phys. Rev. B 48, 7438 (1993). https://doi.org/10.1103/PhysRevB.48.7438

    Article  ADS  Google Scholar 

  14. Knops, Y.M.M., Nienhuis, B., Knops, H.J.F., Blöte, H.W.J.: A 19-vertex version of the fully frustrated XY-model. Phys. Rev. E 50, 1061 (1994)

    Article  ADS  Google Scholar 

  15. Nightingale, M.P., Granato, E., Kosterlitz, J.M.: Conformal anomaly and critical exponents of the XY Ising model. Phys. Rev. B 52, 7402 (1995). https://doi.org/10.1103/PhysRevB.52.7402

    Article  ADS  Google Scholar 

  16. Nicolaides, D.B.: Monte Carlo simulation of the fully frustrated XY model. J. Phys. A 24, L231 (1991)

    Article  ADS  Google Scholar 

  17. Ramirez-Santiago, G., José, J.V.: Correlation functions in the fully Frustrated 2D XY model. Phys. Rev. Lett. 68, 1224 (1992). https://doi.org/10.1103/PhysRevLett.68.1224

    Article  ADS  Google Scholar 

  18. Granato, E., Kosterlitz, J.M., Lee, J., Nightingale, M.P.: Phase transitions in coupled XY-Ising systems. Phys. Rev. Lett. 66, 1090 (1991). https://doi.org/10.1103/PhysRevLett.66.1090

    Article  ADS  Google Scholar 

  19. Grest, G.S.: Critical behavior of the two-dimensional uniformly frustrated charged Coulomb gas. Phys. Rev. B 39, 9267 (1989). https://doi.org/10.1103/PhysRevB.39.9267

    Article  ADS  Google Scholar 

  20. Lee, J.-R.: Phase transitions in the two-dimensional classical lattice Coulomb gas of half-integer charges. Phys. Rev. B 49, 3317 (1994). https://doi.org/10.1103/PhysRevB.49.3317

    Article  ADS  Google Scholar 

  21. Lee, S., Lee, K.-C.: Phase transitions in the fully frustrated XY model studied with use of the microcanonical Monte Carlo technique. Phys. Rev. B 49, 184 (1994). https://doi.org/10.1103/PhysRevB.49.15184

    Article  Google Scholar 

  22. Olsson, P.: Two phase transitions in the fully frustrated XY model. Phys. Rev. Lett. 75, 2758 (1995). https://doi.org/10.1103/PhysRevLett.75.2758

    Article  ADS  Google Scholar 

  23. Granato, E., Kosterlitz, J.M., Nightingale, M.P.: Critical behavior of Josephson-junction arrays at \(f=1/2\). Physica B 222, 266 (1996). https://doi.org/10.1016/0921-4526(96)00204-9

    Article  ADS  Google Scholar 

  24. José, J.V., Ramirez-Santiago, G.: Comment of two phase transitions in the fully frustrated XY model. Phys. Rev. Lett. 77, 4849 (1996). https://doi.org/10.1103/PhysRevLett.77.4849

    Article  ADS  Google Scholar 

  25. Hasenbusch, M., Pelissetto, A., Vicari, E.: Multicritical behaviour in the fully frustrated XY model and related systems. J. Stat. Mech. (2005). https://doi.org/10.1088/1742-5468/2005/12/P12002

  26. Lima, A.B., Costa, B.V.: The Z(2) phase transition in the fully frustrated XY model as a percolation problem. J. Magn. Magn. Mater. 300, 427 (2006). https://doi.org/10.1016/j.jmmm.2005.05.035

    Article  ADS  Google Scholar 

  27. Onsager, L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117–149 (1944). https://doi.org/10.1103/PhysRev.65.117

    Article  MathSciNet  MATH  ADS  Google Scholar 

  28. Landau, D.P., Binder, K. (eds.): A Guid to Monte Carlo Simulations in Statistical Physics, 4th edn. Cambridge Press, Cambridge (2015)

    Google Scholar 

  29. Costa, B.V., Mól, L.A.S., Rocha, J.C.S.: Energy probability distribution zeros: a route to study phase transitions. Comput. Phys. Commun. 216, 77 (2017). https://doi.org/10.1016/j.cpc.2017.03.003

    Article  ADS  Google Scholar 

  30. Costa, B.V., Mól, L.A.S., Rocha, J.C.S.: The zeros of the energy probability distribution—a new way to study phase transitions. J. Phys. Conf. Ser. 921, 1 (2017). https://doi.org/10.1088/1742-6596/921/1/012004

    Article  Google Scholar 

  31. Fisher, M.E.: Renormalization group theory: its basis and formulation in statistical physics. Rev. Mod. Phys. 70, 653–681 (1998). https://doi.org/10.1103/RevModPhys.70.653

    Article  MathSciNet  MATH  ADS  Google Scholar 

  32. Yang, C.N., Lee, T.D.: Statistical theory of equations of state and phase transitions. I. Theory of condensation. Phys. Rev. 87, 404–409 (1952). https://doi.org/10.1103/PhysRev.87.404

    Article  MathSciNet  MATH  ADS  Google Scholar 

  33. Fisher, M.E.: In: Brittin, W. (ed.) Lectures in Theoretical Physics: Volume VII C—Statistical Physics, Weak Interactions, Field Theory: Lectures Delivered at the Summer Institute for Theoretical Physics, University of Colorado, Boulder, 1964, no, vol. 7. University of Colorado Press, Boulder (1965)

  34. Rocha, J.C.S., Schnabel, S., Landau, D.P., Bachmann, M.: Identifying transitions in finite systems by means of partition function zeros and microcanonical inflection-point analysis: a comparison for elastic flexible polymers. Phys. Rev. E 90, 022601 (2014)

    Article  ADS  Google Scholar 

  35. Salsburg, Z.W., Jackson, J.D., Fickett, W., Wood, W.W.: Application of the Monte Carlo method to the lattice gas model. I. Two dimensional triangular lattice. J. Chem. Phys. 30, 65 (1959)

    Article  ADS  Google Scholar 

  36. Ferrenberg, A.M., Swendsen, R.H.: New Monte Carlo technique for studying phase transitions. Phys. Rev. Lett. 61, 2635–2638 (1988). https://doi.org/10.1103/PhysRevLett.61.2635

    Article  ADS  Google Scholar 

  37. Ferrenberg, A.M., Swendsen, R.H.: Optimized Monte Carlo data analysis. Phys. Rev. Lett. 63, 1195–1198 (1989). https://doi.org/10.1103/PhysRevLett.63.1195

    Article  ADS  Google Scholar 

  38. Ferrenberg, A.M., Laudau, D.P., Wong, Y.J.: Monte Carlo simulations: hidden errors from ‘good’ random number generators. Phys. Rev. Lett. 23, 3382 (1992). https://doi.org/10.1103/PhysRevLett.69.3382

    Article  ADS  Google Scholar 

  39. Resende, F.J., Costa, B.V.: Using random number generators in Monte Carlo simulations. Phys. Rev. E 58, 5183 (1998). https://doi.org/10.1103/PhysRevE.58.5183

    Article  ADS  Google Scholar 

  40. Gnu Project. https://gcc.gnu.org/

Download references

Acknowledgements

This work was partially supported by CNPq and Fapemig, Brazilian Agencies.

Author information

Authors and Affiliations

  1. Departamento de Física Aplicada, Universidade Federal do Triângulo Mineiro, Uberaba, Minas Gerais, Brazil

    A. B. Lima

  2. Laboratório de Simulacão, Departamento de Física, ICEx Universidade Federal de Minas Gerais, Belo Horizonte, Minas Gerais, 31720-901, Brazil

    L. A. S. Mól & B. V. Costa

Authors
  1. A. B. Lima
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. A. S. Mól
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. V. Costa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. B. Lima.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lima, A.B., Mól, L.A.S. & Costa, B.V. The Fully Frustrated XY Model Revisited: A New Universality Class. J Stat Phys 175, 960–971 (2019). https://doi.org/10.1007/s10955-019-02271-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-019-02271-x

Search

Navigation