Skip to main content
Log in

Series Analysis of a Kosterlitz-Thouless Transition: The 6-State Planar Potts Model

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

Abstract

A new implementation of the Finite Lattice Method of series expansion is used to derive low temperature series for the 6-state planar Potts model. This is expected to have Kosterlitz-Thouless transitions between the ordered low-temperature phase and a massless phase, and between the massless phase and a high-temperature disordered phase. In an exploratory study, we have analysed series for the order parameter to order x 59. These series have proved particularly difficult to analyse and when seeking to both locate the transition and confirm the form of the Kosterlitz-Thouless transition, considerable ambiguity is found. If however the form of the Kosterlitz-Thouless transition is assumed, then the series analysis consistently indicates that the lower transition occurs at x L=0.4886(10). A more conservative analysis, which made no assumption about the nature of the transition, led to the estimate x L=0.485(8).

This is a preview of subscription content, log in via an institution to check access.

Access this article

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. Baek, S.K., Minnhagen, P.: Non-Kosterlitz-Thouless transitions for the q-state clock models. Phys. Rev. E 82, 031102 (2010)

    Article  ADS  Google Scholar 

  2. Baek, S.K., Minnhagen, P., Beom, J.K.: Comment on “Six-state clock model on the square lattice: Fisher zero approach with Wang-Landau sampling”. Phys. Rev. E 81, 063101 (2010)

    Article  ADS  Google Scholar 

  3. Barber, M., Enting, I.G.: High-field polynomial expansions for the six-state planar Potts model. Aust. J. Phys. 34, 551–561 (1981)

    ADS  Google Scholar 

  4. Baxter, R.J.: Potts model at the critical temperature. J. Phys. C, Solid State Phys. 6, L445 (1973)

    Article  ADS  Google Scholar 

  5. Berezinsky, V.L.: Destruction of long range order in one-dimensional and two-dimensional systems having a continuous symmetry group. 1. Classical systems. Sov. Phys. JETP 32, 493–500 (1971)

    ADS  Google Scholar 

  6. Blöte, H.W.J., Nightingale, M.P.: Critical behaviour of the two-dimensional Potts model with a continuous number of states; A finite size scaling analysis. Physica 112A, 405–465 (1982)

    ADS  Google Scholar 

  7. Briggs, K.M., Enting, I.G., Guttmann, A.J.: Series studies of the Potts model. II. Bulk series for the square lattice. J. Phys. A, Math. Gen. 27, 1503 (1994)

    Article  ADS  MATH  Google Scholar 

  8. Brito, A.F., Redinz, J.A., Plascak, J.A.: Two-dimensional XY and clock models studied via the dynamics generated by rough surfaces. Phys. Rev. E 81, 031130 (2010)

    Article  ADS  Google Scholar 

  9. Cardy, J.L.: General discrete planar models in two dimensions: duality properties and phase diagrams. J. Phys. A, Math. Gen. 13, 1507 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  10. Domany, E., Mukamel, D., Schwimmer, A.: Phase diagram of the Z(5) model on a square lattice. J. Phys. A, Math. Gen. 13, L311 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  11. Domb, C.: Configurational studies of the Potts models. J. Phys. A, Math. Nucl. Gen. 7, 1335 (1974)

    Article  ADS  Google Scholar 

  12. Elitzur, S., Pearson, R.B., Shigemitsu, J.: Phase structure of discrete Abelian spin and gauge systems. Phys. Rev. D 19, 3698–3714 (1979)

    Article  ADS  Google Scholar 

  13. Enting, I.G.: Generalised Möbius functions for rectangles on the square lattice. J. Phys. A, Math. Gen. 11, 563–568 (1978)

    Article  MathSciNet  ADS  Google Scholar 

  14. Enting, I.G.: Generating functions for enumerating self-avoiding rings on the square lattice. J. Phys. A, Math. Gen. 13, 3713–3722 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  15. Enting, I.G.: Universality in a generalised Potts model. J. Phys. A, Math. Gen. 13, L409 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  16. Enting, I.G.: Series expansions from the finite lattice method. Nucl. Phys. B, Proc. Suppl. 47, 180–187 (1996)

    Article  ADS  Google Scholar 

  17. Enting, I.G.: Lattice statistics studies of massless phases. In: Abbott, D., Batchelor, M., Dewar, R., di Matteo, T., Guttmann, T. (eds.) Complex Systems II. Proceedings of SPIE, vol. 6802 (2008)

    Google Scholar 

  18. Enting, I.G., Domb, C.: Cayley tree approximation for the Potts model. J. Phys. A, Math. Gen. 8, 1228–1235 (1975)

    Article  ADS  Google Scholar 

  19. Fortuin, C.M., Kasteleyn, P.W.: On the random-cluster model I. Introduction and relation to other models. Physica 57, 536–564 (1972)

    Article  MathSciNet  ADS  Google Scholar 

  20. Gaunt, D.S., Guttmann, A.J.: Asymptotic analysis of coefficients. In: Domb, C., Green, M.S. (eds.) Phase Transitions and Critical Phenomena, vol. 3. Academic Press, New York (1974)

    Google Scholar 

  21. Guttmann, A.J.: Asymptotic analysis of power-series expansions. In: Domb, C., Lebowitz, J.L. (eds.) Phase Transitions and Critical Phenomena, vol. 13, pp. 1–234. Academic Press, New York (1989)

    Google Scholar 

  22. Guttmann, A.J., Enting, I.G.: The number of convex polygons on the square and honeycomb lattices. J. Phys. A, Math. Gen. 21, L467 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  23. Guttmann, A.J., Enting, I.G.: The phase transition of the 3-dimensional 3-state Potts model. Nucl. Phys. B, Proc. Suppl. 17, 328–330 (1990)

    Article  ADS  Google Scholar 

  24. Guttmann, A.J., Enting, I.G.: Series studies of the Potts model: III. The 3-state model on the simple cubic lattice. J. Phys. A, Math. Gen. 27, 5801 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Hwang, C.-O.: Six-state clock model on the square lattice: Fisher zero approach with Wang-Landau sampling. Phys. Rev. E 80, 042103 (2009)

    Article  ADS  Google Scholar 

  26. Jensen, I.: Enumeration of self-avoiding walks on the square lattice. J. Phys. A, Math. Gen. 37, 5503–5524 (2004)

    Article  ADS  MATH  Google Scholar 

  27. Jensen, I., Guttmann, A.J.: Self-avoiding polygons on the square lattice. J. Phys. A, Math. Gen. 32, 4867–4876 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. José, J.V., Kadanoff, L.P., Kirkpatrick, S., Nelson, D.R.: Renormalization, vortices and symmetry-breaking perturbations in the two-dimensional planar model. Phys. Rev. B 16, 1217–1241 (1977)

    Article  ADS  Google Scholar 

  29. Kenna, R., Irving, A.C.: The Kosterlitz-Thouless universality class. Nucl. Phys. B 485, 583–612 (1997)

    Article  ADS  Google Scholar 

  30. Kosterlitz, J.M.: The critical properties of the two-dimensional xy model. J. Phys. C, Solid State Phys. 7, 1046 (1974)

    Article  ADS  Google Scholar 

  31. 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 

  32. Lapilli, C.M., Pfeifer, P., Wexler, C.: Universality away from critical points in two-dimensional phase transitions. Phys. Rev. Lett. 96, 140603 (2006)

    Article  ADS  Google Scholar 

  33. Murty, S., Challa, S., Landau, D.P.: Critical behavior of the six-state clock model in two dimensions. Phys. Rev. B 33, 437–443 (1986)

    Article  ADS  Google Scholar 

  34. Nienhuis, B., Berker, A.N., Riedel, E.K., Schick, M.: First- and second-order phase transitions in Potts models: renormalization-group solution. Phys. Rev. Lett. 43, 737–740 (1979)

    Article  ADS  Google Scholar 

  35. Potts, R.B.: Some generalised order-disorder transformations. Proc. Camb. Philol. Soc. 48, 106–109 (1952)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. Salvy, B.: Examples of automatic asymptotic expansions. SIGSAM Bull. 25, 4–17 (1991)

    Article  Google Scholar 

  37. Tobochnik, J.: Properties of the q-state clock model for q=4,5, and 6. Phys. Rev. B 26, 6201–6207 (1982)

    Article  ADS  Google Scholar 

  38. Tomita, Y., Okabe, Y.: Probability-changing cluster algorithm for two-dimensional XY and clock models. Phys. Rev. B 65, 184405 (2002)

    Article  ADS  Google Scholar 

  39. Tversky, A., Kahneman, D.: Judgment under uncertainty: Heuristics and biases. Science 185, 1124–1131 (1974)

    Article  ADS  Google Scholar 

  40. Wu, F.Y.: Phase diagram of a five-state spin system. J. Phys. C, Solid State Phys. 12, L317 (1979)

    Article  ADS  Google Scholar 

  41. Wu, F.Y.: The Potts model. Rev. Mod. Phys. 54, 235–268 (1982)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to N. Clisby.

Additional information

To mark the 90th birthday of Professor Cyril Domb.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Enting, I.G., Clisby, N. Series Analysis of a Kosterlitz-Thouless Transition: The 6-State Planar Potts Model. J Stat Phys 145, 696–712 (2011). https://doi.org/10.1007/s10955-011-0322-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-011-0322-8

Keywords

Navigation