Ground-state degeneracy of Potts antiferromagnets on two-dimensional lattices: Approach using infinite cyclic strip graphs

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 60, Issue 4 A, 1999, Pages 3512-3515

  Shrock, Robert ^a   Tsai, Shan Ho ^b  

^a STONY BROOK UNIVERSITY   (United States)

^b UNIVERSITY OF GEORGIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000793230     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/physreve.60.3512     Document Type: Article

References (31)

1. M. Aizenman and E. H. Lieb, J. Stat. Phys. 24, 279 (1981); Y. Chow and F. Y. Wu, Phys. Rev. B 36, 285 (1987).
2. M. Aizenman and E. H. Lieb, J. Stat. Phys. 24, 279 (1981); Y. Chow and F. Y. Wu, Phys. Rev. B 36, 285 (1987).
3. R. B. Potts, Proc. Cambridge Philos. Soc. 48, 106 (1952).
4. F. Y. Wu, Rev. Mod. Phys. 54, 235 (1982); 55, 315(E) (1983).
5. F. Y. Wu, Rev. Mod. Phys. 54, 235 (1982); 55, 315(E) (1983).
6. R. C. Read, J. Comb. Theory, Ser. A 4, 52 (1968); R. C. Read and W. T. Tutte, in Selected Topics in Graph Theory, 3, edited by L. W. Beineke and R. J. Wilson (Academic Press, New York, 1988).
7. R. C. Read, J. Comb. Theory, Ser. A 4, 52 (1968); R. C. Read and W. T. Tutte, in Selected Topics in Graph Theory, 3, edited by L. W. Beineke and R. J. Wilson (Academic Press, New York, 1988).
8. The minimum number of colors needed for this coloring of G is called its chromatic number, χ(G).
9. s) [7]. As in Ref. [7], we shall use the first order of limits here; this has the advantage of removing certain isolated discontinuities in W.
10. R. Shrock and S.-H. Tsai, Phys. Rev. E 55, 5165 (1997).
11. R. J. Baxter, J. Phys. A 20, 5241 (1987).
12. R. Shrock and S.-H. Tsai, Phys. Rev. E 56, 1342 (1997).
13. M. Roček, R. Shrock, and S.-H. Tsai, Physica A 252, 505 (1998); 259, 367 (1998).
14. M. Roček, R. Shrock, and S.-H. Tsai, Physica A 252, 505 (1998); 259, 367 (1998).
15. R. Shrock and S.-H. Tsai, Phys. Rev. E 56, 3935 (1997); 56, 4111 (1997).
16. R. Shrock and S.-H. Tsai, Phys. Rev. E 56, 3935 (1997); 56, 4111 (1997).
17. R. Shrock and S.-H. Tsai, J. Phys. A 31, 9641 (1998).
18. R. Shrock and S.-H. Tsai, Physica A 259, 315 (1998).
19. R. Shrock and S.-H. Tsai, Phys. Rev. E 58, 4332 (1998); e-print cond-mat/9808057. In the absence of exact solutions, these values of W(Λ,q) are determined by Monte Carlo measurements and large-q series.
20. R. Shrock and S.-H. Tsai, Physica A 265, 186 (1999); J. Phys. A 32, L195 (1999).
21. R. Shrock and S.-H. Tsai, Physica A 265, 186 (1999); J. Phys. A 32, L195 (1999).
22. E. H. Lieb, Phys. Rev. 162, 162 (1967).
23. R. J. Baxter, J. Math. Phys. 11, 784 (1970).
24. R. Shrock and S.-H. Tsai, J. Phys. A 30, 495 (1997).
25. N. L. Biggs, R. M. Damerell, and D. A. Sands, J. Comb. Theory, Ser. B 12, 123 (1972); N. L. Biggs and G. H. Meredith, ibid. 20, 5 (1976).
26. N. L. Biggs, R. M. Damerell, and D. A. Sands, J. Comb. Theory, Ser. B 12, 123 (1972); N. L. Biggs and G. H. Meredith, ibid. 20, 5 (1976).
27. S. Beraha, J. Kahane, and N. Weiss, J. Comb. Theory, Ser. B 28, 52 (1980).
28. R. C. Read and G. F. Royle, in Graph Theory, Combinatorics, and Applications (Wiley, New York, 1991), Vol. 2, p. 1009.
29. Some families of graphs that do not have regular lattice directions have noncompact loci B that separate the q plane into different regions [11,12,15].
30. 2. Graph families with B not including q=0 are given in [9,11,12,15].
31. c({G}), one can only determine the magnitude |W({G},q)| unambiguously [7].