Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T01:42:14.006Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Ashkin–Teller Model and Tutte–Whitney Functions

Published online by Cambridge University Press:  01 March 2007

G. E. FARR
G. E. FARR*
Affiliation:
Clayton School of Information Technology, Monash University, Clayton, Victoria 3800, Australia (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

The partition functions of the Ising and Potts models in statistical mechanics are well known to be partial evaluations of the Tutte–Whitney polynomial of the appropriate graph. The Ashkin–Teller model generalizes the Ising model and the four-state Potts model, and has been extensively studied since its introduction in 1943. However, its partition function (even in the symmetric case) is not a partial evaluation of the Tutte–Whitney polynomial. In this paper, we show that the symmetric Ashkin–Teller partition function can be obtained from a generalized Tutte–Whitney function which is intermediate in a precise sense between the usual Tutte–Whitney polynomialof the graph and that of its dual.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 16 , Issue 2 , March 2007 , pp. 251 - 260
Copyright
Copyright © Cambridge University Press 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ashkin, J. and Teller, E. (1943) Statistics of two-dimensional lattices with four components. Phys. Rev. 64 178–184.CrossRefGoogle Scholar
[2]Baxter, R. J. (1982) Exactly Solved Models in Statistical Mechanics, Academic Press, London.Google Scholar
[3]Brylawski, T. H. and Oxley, J. G. (1992) The Tutte polynomial and its applications. In Matroid Applications (White, N., ed.), Vol. 40 of Encyclopedia Math. Appl., Cambridge University Press, pp. 123–225.CrossRefGoogle Scholar
[4]Chayes, L. and Machta, J. (1997) Graphical representation and cluster algorithms I: Discrete spin systems. Physica A 239 542–601.CrossRefGoogle Scholar
[5]Edwards, R. G. and Sokal, A. D. (1988) Generalization of the Fortuin–Kasteleyn–Swendsen–Wang representation and Monte-Carlo algorithm. Phys. Rev. D 38 2009–2012.CrossRefGoogle ScholarPubMed
[6]Farr, G. E. (1993) A generalization of the Whitney rank generating function. Math. Proc. Camb. Phil. Soc. 113 267–280.CrossRefGoogle Scholar
[7]Farr, G. E. (2004) Some results on generalised Whitney functions. Adv. Appl. Math. 32 239–262.CrossRefGoogle Scholar
[8]Farr, G. E. Tutte–Whitney polynomials: Some history and generalizations. In Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh (G. R. Grimmett and C. J. H. McDiarmid, eds), Oxford University Press, to appear.Google Scholar
[9]Fortuin, C. M. and Kasteleyn, P. W. (1972) On the random cluster model I: Introduction and relation to other models. Physica 57 536–564.CrossRefGoogle Scholar
[10]Greene, C. (1976) Weight enumeration and the geometry of linear codes. Stud. Appl. Math. 55 119–128.CrossRefGoogle Scholar
[11]Ising, E. (1925) Beitrag zur Theorie des Ferromagnetismus. Z. Phys. 31 253–258.CrossRefGoogle Scholar
[12]Jaeger, F., Vertigan, D. L. and Welsh, D. J. A. (1990) On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Camb. Phil. Soc. 108 35–53.CrossRefGoogle Scholar
[13]Jerrum, M. and Sinclair, A. (1993) Polynomial-time approximation algorithms for the Ising model. SIAM J. Comput. 22 1087–1116.CrossRefGoogle Scholar
[14]Kasteleyn, P. W. and Fortuin, C. M. (1969) Phase transitions in lattice systems with local properties. J. Phys. Soc. Japan Suppl. 26 11–14.Google Scholar
[15]Kung, J. P. S. (1980) The Rédei function of a relation. J. Combin. Theory Ser. A 29 287–296.CrossRefGoogle Scholar
[16]Pfister, C.-E. and Velenik, Y. (1997) Random-cluster representation of the Ashkin–Teller model. J. Statist. Phys. 88 1295–1331.CrossRefGoogle Scholar
[17]Potts, R. B. (1952) Some generalized order-disorder transformations. Proc. Camb. Phil. Soc. 48 106–109.CrossRefGoogle Scholar
[18]Salas, J. and Sokal, A. D. (1996) Dynamic critical behavior of a Swendsen–Wang-type algorithm for the Ashkin–Teller model. J. Statist. Phys. 85 297–361.CrossRefGoogle Scholar
[19]Sokal, A. D. (2005) The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In Surveys in Combinatorics, 2005 (Webb, B. S., ed.), Vol. 327 of London Math. Soc. Lecture Note Series, Cambridge University Press, pp. 173–226.CrossRefGoogle Scholar
[20]Swendsen, R. H. and Wang, J.-S. (1987) Nonuniversal critical dynamics in Monte-Carlo simulations. Phys. Rev. Lett. 58 86–88.CrossRefGoogle ScholarPubMed
[21]Tutte, W. T. (1947) A ring in graph theory. Proc. Camb. Phil. Soc. 43 26–40.CrossRefGoogle Scholar
[22]Tutte, W. T. (1954) A contribution to the theory of chromatic polynomials. Canadian J. Math. 6 80–91.CrossRefGoogle Scholar
[23]Tutte, W. T. (1974) Codichromatic graphs. J. Combin. Theory Ser. B 16 168–174.CrossRefGoogle Scholar
[24]Welsh, D. J. A. (1976) Matroid Theory, Vol. 8 of London Math. Soc. Monographs, Academic Press, London.Google Scholar
[25]Welsh, D. J. A. (1993) Complexity: Knots, Colourings and Counting, Vol. 186 of London Math. Soc. Lecture Note Series, Cambridge University Press.CrossRefGoogle Scholar
[26]Whitney, H. (1932) The coloring of graphs. Ann. of Math. (2) 33 688–718.CrossRefGoogle Scholar
[27]Wu, F. Y. (1977) Ashkin–Teller model as a vertex problem. J. Math. Phys. 18 611–613.CrossRefGoogle Scholar
[28]Wu, F. Y. (1982) The Potts model. Rev. Mod. Phys. 54 235–268.CrossRefGoogle Scholar
[29]Wu, F. Y. and Wang, Y. K. (1976) Duality transformation in a many-component spin model. J. Math. Phys. 17 439–440.CrossRefGoogle Scholar