Skip to main content
SpringerLink
Log in

A New Algorithm to Study the Critical Behavior of Topological Phase Transitions

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

Abstract

Topological phase transitions such as the Berezinskii-Kosterlitz-Thouless (BKT) transition are difficult to characterize due to the difficulty in defining an appropriate order parameter or to unravel its critical properties. In this paper, we discuss the application of a newly introduced numerical algorithm that was inspired by the Fisher zeros of the partition function and is based on the partial knowledge of the zeros of the energy probability distribution (EPD zeros). This iterative method has proven to be quite general, furnishing the transition temperature with great precision and a relatively low computational effort. Since it does not need the a priori knowledge of any order parameter it provides an unbiased estimative of the transition temperature being convenient to the study of this kind of phase transition. Therefore, we applied the EPD zeros approach to the 2D XY model, which is well known for showing a BKT transition, in order to demonstrate its effectivity in the study of the BKT transition. Our results are consistent with the real and imaginary parts of the pseudo-transition temperature, T(L), having a different asymptotic behavior, which suggests a way to characterize a BKT like transition.

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

Similar content being viewed by others

References

  1. N.D. Mermin, H. Wagner, 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. 40 Years History of Berezinskii-Kosterlitz-Thouless Theory, edited by Jorge V. José (World Scientific) (2013)

  3. P. Minnhagen, The two-dimensional Coulomb gas, vortex unbinding, and superfluid-superconducting films. Rev. Modern Phys. 59, 1001 (1987). https://doi.org/10.1103/RevModPhys.59.1001

    Article  ADS  Google Scholar 

  4. V.L. Berezinskii, 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 

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

    Article  ADS  Google Scholar 

  6. L. Pitaevskii, S. Stringari, Bose–Einstein condensation and superfluidity, international series of monographs on physics, Oxford University Press (2016)

  7. R.J. Fletcher, Bose-Einstein condensation and superfluidity in two dimensions, Dissertation submitted for the degree of Doctor of Philosophy, Trinity College, University of Cambridge (2015)

  8. K. Gretchen, Campbell, Superfluidity goes 2D. Nat. Phys. 8, 643 (2012). https://doi.org/10.1038/nphys2395

    Article  Google Scholar 

  9. D.C. Mattis, Statistical mechanics made simple : a guide for students and researchers. World Scientific Publishing (2003)

  10. J.C.S. Rocha, B.V. Costa, L.A.S. Mól, Using zeros of the canonical partition function map to detect signatures of a Berezinskii-Kosterlitz-Thouless transition. Comput. Phys. Commun. 209, 88–91 (2016). https://doi.org/10.1016/j.cpc.2016.08.016

    Article  MathSciNet  MATH  ADS  Google Scholar 

  11. T.P. Figueiredo, J.C.S. Rocha, B.V. Costa, Topological phase transition in the two-dimensional anisotropic Heisenberg model: a study using the replica exchange wang-landau sampling. Physica A-Statistical Mechanics and its Applications. 488, 121 (2017). https://doi.org/10.1016/j.physa.2017.07.010

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  13. H.G. Evertz, D.P. Landau, Critical dynamics in the 2D classical XY-model: a spin dynamics study. Phys. Rev. B. 54, 12302 (1996). https://doi.org/10.1103/PhysRevB.54.12302

    Article  ADS  Google Scholar 

  14. B.V. Costa, P.Z. Coura, S.A. Leonel, Berezinskii–kosterlitz–thouless transition close to the percolation threshold. Phys. Lett. A. 377, 1239 (2013). https://doi.org/10.1016/j.physleta.2013.03.030

    Article  MATH  ADS  Google Scholar 

  15. M.E. Fisher, 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 

  16. C.N. Yang, T.D. Lee, 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 

  17. M.E. Fisher, in . 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. v. 7, University of Colorado Press, Boulder, ed. by W. Brittin, (1965)

  18. B.V. Costa, L.A.S. Mól, J.C.S. Rocha, 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 

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

    Article  Google Scholar 

  20. T. Vogel, Y.W. Li, T. Wüst, D.P. Landau, Generic, hierarchical framework for massively parallel wang-landau sampling. Phys. Rev. Lett. 110, 210603 (2013). https://doi.org/10.1103/PhysRevLett.110.210603

    Article  ADS  Google Scholar 

  21. T. Vogel, Y.W. Li, T. Wüst, D.P. Landau, Scalable replica-exchange framework for Wang-Landau sampling. Phys. Rev. E. 90, 023302 (2014). https://doi.org/10.1103/PhysRevE.90.023302

    Article  ADS  Google Scholar 

  22. F. Wang, D.P. Landau, Efficient, Multiple-Range random walk algorithm to calculate the density of states. Phys. Rev. Lett. 86, 2050–2053 (2001). https://doi.org/10.1103/PhysRevLett.86.2050

    Article  ADS  Google Scholar 

  23. A.M. Ferrenberg, R.H. Swendsen, 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 

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

    Article  ADS  Google Scholar 

Download references

Funding

The authors thank CNPq (Grants 303480/2017-3, 306457/2016-4) and FAPEMIG (APQ-03183-16) for the partial financial support.

Author information

Authors and Affiliations

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

    B. V. Costa & J. C. S. Rocha

  2. Departamento de Física, ICEB, Universidade Federal de Ouro Preto, 35400-000, Ouro Preto, Minas Gerais, Brazil

    L. A. S. Mól

Authors
  1. B. V. Costa
    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. J. C. S. Rocha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. V. Costa.

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

Costa, B.V., Mól, L.A.S. & Rocha, J.C.S. A New Algorithm to Study the Critical Behavior of Topological Phase Transitions. Braz J Phys 49, 271–276 (2019). https://doi.org/10.1007/s13538-019-00636-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13538-019-00636-x

Keywords

Search

Navigation