Journal of the Korean Physical Society

, Volume 72, Issue 6, pp 646–652 | Cite as

Partition Function Zeros and Shift Exponent of the Ising Model on a Square Lattice with Self-dual Boundary Conditions

  • Seung-Yeon Kim


The exact integer values for the density of states of the Ising model on an L×L square lattice with self-dual boundary conditions are evaluated up to L = 22 for the first time by counting all possible spin states (enormous 2485 ≈ 10146 states for L = 22). The exact partition function zeros in the complex temperature plane, very sensitive indicators for phase transitions and critical phenomena, are obtained by using the density of states. From the behavior of the partition function zeros, the shift exponent λ of the square-lattice Ising model with self-dual boundary conditions is accurately and precisely estimated to be λ = 2.0000000(2), clearly indicating λ = 2.


Self-dual boundary conditions Partition function zeros Shift exponent 


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  1. [1]
    C. Domb, The Critical Point (Taylor and Francis, London, 1996).Google Scholar
  2. [2]
    J. L. Cardy, Finite-Size Scaling (North Holland, Amsterdam, 1988).Google Scholar
  3. [3]
    M. N. Barber, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic Press, New York, 1983), Vol. 8, p. 145.Google Scholar
  4. [4]
    A. E. Ferdinand and M. E. Fisher, Phys. Rev. 185, 832 (1969).ADSCrossRefGoogle Scholar
  5. [5]
    C. Hoelbling and C. B. Lang, Phys. Rev. B 54, 3434 (1996).ADSCrossRefGoogle Scholar
  6. [6]
    W. Janke and R. Kenna, Phys. Rev. B 65, 064110 (2002).ADSCrossRefGoogle Scholar
  7. [7]
    S-Y. Kim, J. Korean Phys. Soc. 65, 2009 (2014).ADSCrossRefGoogle Scholar
  8. [8]
    A. Poghosyan, R. Kenna and N. Izmailian, Europhys. Lett. 111, 60010 (2015).ADSCrossRefGoogle Scholar
  9. [9]
    R. J. Creswick, Phys. Rev. E 52, R5735 (5735).Google Scholar
  10. [10]
    S-Y. Kim and R. J. Creswick, Phys. Rev. Lett. 81, 2000 (1998).ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    S-Y. Kim, Nucl. Phys. B 637, 409 (2002).ADSCrossRefGoogle Scholar
  12. [12]
    S-Y. Kim, Phys. Rev. Lett. 93, 130604 (2004).ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    S-Y. Kim, J. Korean Phys. Soc. 67, 1517 (2015).ADSCrossRefGoogle Scholar
  14. [14]
    C. N. Yang and T. D. Lee, Phys. Rev. 87, 404 (1952).ADSCrossRefGoogle Scholar
  15. [14a]
    T. D. Lee and C. N. Yang, ibid. 410 (1952).Google Scholar
  16. [15]
    M. E. Fisher, in Lectures in Theoretical Physics, edited by W. E. Brittin (University of Colorado Press, Boulder, CO, 1965), Vol. 7c, p. 1.Google Scholar
  17. [16]
    C. Itzykson, R. B. Pearson and J-B. Zuber, Nucl. Phys. B 220, 415 (1983).ADSCrossRefGoogle Scholar
  18. [17]
    W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in Fortran 77, 2nd edition (Cambridge University Press, Cambridge, 1992), p. 104.zbMATHGoogle Scholar
  19. [18]
    W. T. Lu and F. Y. Wu, Physica A 258, 157 (1998).ADSMathSciNetCrossRefGoogle Scholar
  20. [19]
    I. Bena, M. Droz and A. Lipowski, Int. J. Mod. Phys. B 19, 4269 (2005). and references therein.ADSCrossRefGoogle Scholar
  21. [20]
    S-Y. Kim, C-O. Hwang and J. M. Kim, Nucl. Phys. B 805, 441 (2008).ADSCrossRefGoogle Scholar
  22. [21]
    J. H. Lee, S-Y. Kim and J. Lee, J. Chem. Phys. 133, 114106 (2010).ADSCrossRefGoogle Scholar
  23. [22]
    J. H. Lee, S-Y. Kim and J. Lee, Phys. Rev. E 86, 011802 (2012).ADSCrossRefGoogle Scholar
  24. [23]
    J. Lee, Phys. Rev. Lett. 110, 248101 (2013).ADSCrossRefGoogle Scholar
  25. [24]
    S-Y. Kim and W. Kwak, J. Korean Phys. Soc. 65, 436 (2014).ADSCrossRefGoogle Scholar
  26. [25]
    J. H. Lee, S-Y. Kim and J. Lee, AIP Advances 5, 127211 (2015).ADSCrossRefGoogle Scholar

Copyright information

© The Korean Physical Society 2018

Authors and Affiliations

  1. 1.School of Liberal Arts and SciencesKorea National University of TransportationChungjuKorea

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