Skip to main content
Log in

Density of the Fisher Zeroes for the Ising Model

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

Abstract

The density of the Fisher zeroes, or zeroes of the partition function in the complex temperature plane, is determined for the Ising model in zero field as well as in a pure imaginary field iπ/2. Results are given for the simple-quartic, triangular, honeycomb, and the kagomé lattices. It is found that the density diverges logarithmically at points along its loci in appropriate variables.

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. C. N. Yang and T. D. Lee, Statistical theory of equations of states and phase transitions I. Theory of condensation, Phys. Rev. 87: 404–409 (1952).

    Google Scholar 

  2. T. D. Lee and C. N. Yang, Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model, Phys. Rev. 87: 410–419 (1952).

    Google Scholar 

  3. M. E. Fisher, The Nature of critical points, in Lecture Notes in Theoretical Physics, Vol. 7c, W. E. Brittin, ed. (University of Colorado Press, Boulder, 1965), pp. 1–159.

    Google Scholar 

  4. J. Stephenson and R. Couzens, Partition function zeros for the two-dimensional Ising model, Physica A 129: 201–210 (1984).

    Google Scholar 

  5. W. T. Lu and F. Y. Wu, Partition function zeroes of a self-dual Ising model, Physica A 258: 157–170 (1998).

    Google Scholar 

  6. J. Stephenson, Partition function zeros for the two-dimensional Ising model. II, Physica A 136: 147–159 (1986).

    Google Scholar 

  7. J. Stephenson, On the density of partition function zeros, J. Phys. A 20: 4513–4519 (1987).

    Google Scholar 

  8. H. J. Brascamp and H. Kunz, Zeroes of the partition function for the Ising model in the complex temperature plane, J. Math. Phys. 15: 65–66 (1974).

    Google Scholar 

  9. C. N. Chen, C. K. Hu, and F. Y. Wu, Partition function zeros of the square lattice Potts model, Phys. Rev. Lett. 76: 169–172 (1996).

    Google Scholar 

  10. R. M. F. Houtappel, Order_disorder in hexagonal lattice, Physica 16: 425–455 (1950).

    Google Scholar 

  11. G. H. Wannier, Antiferromagnetism: The triangular Ising Net, Phys. Rev. 79: 357–364 (1950).

    Google Scholar 

  12. V. Matveev and R. Shrock, Complex-temperature properties of the 2D Ising model for nonzero magnetic field, Phys. Rev. E 53: 254–267 (1996).

    Google Scholar 

  13. B. M. McCoy and T. T. Wu, Theory of Toeplitz determinants and the spin correlations of the two-dimensional Ising model. II, Phys. Rev. 155: 438–452 (1967).

    Google Scholar 

  14. K. Y. Lin and F. Y. Wu, Ising model in the magnetic field iπkT/2, Int. J. Mod. Phys. B 4: 471–481 (1988).

    Google Scholar 

  15. F. Y. Wu, Two-dimensional Ising model with crossing and four-spin interactions and a magnetic field iπkT/2, J. Stat. Phys. 44: 455–463 (1986).

    Google Scholar 

  16. V. Matveev and R. Shrock, Complex temperature properties of the 2D Ising model with βH=/2, J. Phys. A 28: 4859–4882 (1995).

    Google Scholar 

  17. C. Fan and F. Y. Wu, General lattice model of phase transitions, Phys. Rev. B 2: 723–733 (1970).

    Google Scholar 

  18. C. S. Hsue, K. Y. Lin, and F. Y. Wu, Staggered eight-vertex model, Phys. Rev. B 12: 429–437 (1975).

    Google Scholar 

  19. F. Y. Wu and Y. K. Wang, Duality transformation in a many component spin model, J. Math. Phys. 17: 439–440 (1976).

    Google Scholar 

  20. I. Syozi, Transformation of Ising Models, in Phase Transition and Critical Phenomena, Vol. 1, C. Domb and M. Green, eds. (Academic Press, London, 1970), pp. 269–329.

    Google Scholar 

  21. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 5th ed. (Academic Press, New York, 1994).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lu, W.T., Wu, F.Y. Density of the Fisher Zeroes for the Ising Model. Journal of Statistical Physics 102, 953–970 (2001). https://doi.org/10.1023/A:1004863322373

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1004863322373

Navigation