Abstract
We introduce a new method to generate duality relations for correlation functions of the Potts model on a planar graph. The method extends previously known results, by allowing the consideration of the correlation function for arbitrarily placed vertices on the graph. We show that generally it is linear combinations of correlation functions, not the individual correlations, that are related by dualities. The method is illustrated in several non-trivial cases, and the relation to earlier results is explained. A graph-theoretical formulation of our results in terms of rooted dichromatic, or Tutte, polynomials is also given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
R. B. Potts, Some generalized order-disorder transformations, Proc. Camb. Philos. Soc. 48:106–109 (1952).
R. J. Baxter, Potts model at the critical temperature, J. Phys. C: Solid State Phys. 6:L445–448 (1973).
M. P. M. den Nijs, A relation between the temperature exponents of the 8–vertex and q-state Potts model, J. Phys. A: Math. Gen. 12:1857–1868 (1979).
F. Y. Wu, The Potts model, Rev. Mod. Phys. 54:235–268 (1982).
F. Y. Wu, Potts model of magnetism, J. Appl. Phys. 55:2421–2426 (1984).
F. Y. Wu and Y. K. Wang, Duality transformation in a many component spin model, J. Math. Phys. 17:439–440 (1976).
P. G. Watson, Impurities and defects in Ising lattices, J. Phys. C: Solid State Phys. 1:575–581 (1968).
F. Y. Wu, Duality relation for Potts correlation functions, Phys. Lett. A 228:43–47 (1997).
F. Y. Wu and H. Y. Huang, Sum rule identities and the duality relation for the Potts n-point boundary correlation functions, Phys. Rev. Lett. 79:4954–4957 (1997).
W. T. Lu and F. Y. Wu, On the duality relation for correlation functions of the Potts model, J. Phys. A: Math. Gen. 31:2823–2836 (1998).
C. King, Two-dimensional Potts model and annular partitions, J. Statist. Phys. 96:1071–1089 (1999).
J. L. Jacobsen, Comment on ''Duality relation for Potts correlation functions'' by Wu, Phys. Lett. A 233:489–492 (1997).
F. Y. Wu, C. King, and W. T. Lu, On the rooted Tutte polynomial, Ann. Inst. Fourier, Grenoble 49:1103–1114 (1999).
J. L. Jacobsen and J. Cardy, Critical behaviour of random bond Potts model, Nucl. Phys. B 515[FS]:701–742 (1998).
C. M. Fortuin and P. W. Kasteleyn, On the random-cluster model I. Introduction and relation to other models, Physica 57:536–564 (1972).
I. J. van Lint and R. M. Wilson, A Course in Combinatorics (Cambridge University Press, Cambridge, 1992), p. 301.
R. P. Stanley, Enumerative Combinatorics, Vol. 1 (Wadsworth and Brooks/Cole, Montery, California, 1986).
W. T. Tutte, A contribution to the theory of chromatic polynomials, Can. J. Math. 6:80–91 (1954).
W. T. Tutte, On dichromatic polynomials, J. Comb. Theory 2:301–320 (1967).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
King, C., Wu, F.Y. New Correlation Duality Relations for the Planar Potts Model. Journal of Statistical Physics 107, 919–940 (2002). https://doi.org/10.1023/A:1014550516842
Issue Date:
DOI: https://doi.org/10.1023/A:1014550516842