Lower bounds and series for the ground-state entropy of the Potts antiferromagnet on Archimedean lattices and their duals

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

Volumn 56, Issue 4, 1997, Pages 4111-4124

  Shrock, Robert ^a   Tsai, Shan Ho ^a  

^a STONY BROOK UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001679220     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.56.4111     Document Type: Article

References (54)

1. W. F. Giauque and J. W. Stout, J. Am. Chem. Soc. 58, 1144 (1936).JACSAT
2. L. Pauling, The Nature of the Chemical Bond (Cornell University Press, Ithaca, 1960), p. 466.
3. N. G. Parsonage and L. A. K. Staveley, Disorder in Crystals (Oxford University Press, New York, 1978).
4. P. W. Atkins, Physical Chemistry (Freeman, San Francisco, 1994), p. 710.
5. G. H. Wannier, Phys. Rev. 79, 357 (1950).PHRVAO
6. K. Kano and S. Naya, Prog. Theor. Phys. 10, 158 (1953).PTPKAV
7. R. B. Potts, Proc. Camb. Philos. Soc. 48, 106 (1952).
8. C. M. Fortuin and P. W. Kasteleyn, Physica 57, 536 (1972).PHYSAG
9. F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969).
10. N. L. Biggs, Algebraic Graph Theory, 2nd ed. (Cambridge University Press, Cambridge, 1993).
11. W. T. Tutte, Graph Theory, Vol. 21 of Encyclopedia of Mathematics and its Applications, edited by G. C. Rota (Addison-Wesley, New York, 1984).
12. R. C. Read and W. T. Tutte, in Chromatic Polynomials, edited by L. W. Beineke and R. J. Wilson, Selected Topics in Graph Theory, 3 (Academic, New York, 1988).
13. E. H. Lieb, Phys. Rev. 162, 162 (1967).PHRVAO
14. R. J. Baxter, J. Phys. A 20, 5241 (1987).JPHAC5
15. F. Y. Wu, Rev. Mod. Phys. 54, 235 (1982); RMPHAT
16. R. J. Baxter, J. Math. Phys. (N.Y.) 11, 784 (1970).JMAPAQ
17. R. Shrock and S.-H. Tsai, J. Phys. A 30, 495 (1997).JPHAC5
18. R. Shrock and S.-H. Tsai, Phys. Rev. E 55, 5165 (1997); PLEEE8
19. 55, 5184 (1997).
20. R. Shrock and S.-H. Tsai, Phys. Rev. E 55, 6791 (1997).PLEEE8
21. R. Shrock and S.-H. Tsai, Phys. Rev. E 56, 2733 (1997).PLEEE8
22. R. Shrock and S.-H. Tsai, Phys. Rev. E 56, 3935 (1997).PLEEE8
23. A. P. Ramirez, G. P. Espinosa, and A. S. Cooper, Phys. Rev. Lett. 64, 2070 (1990); PRLTAO
24. C. Broholm, G. Aeppli, G. P. Espinosa, and A. S. Cooper, 65, 3173 (1990).
25. A. S. Wills, A. Harrison, S. A. M. Mentink, T. E. Mason, and Z. Tun, cond-mat/9607106.
26. C. Zeng and V. Elser, Phys. Rev. B 42, 8436 (1990); PRBMDO
27. R. R. P. Singh and D. Huse, Phys. Rev. Lett. 68, 1766 (1992); PRLTAO
28. N. Elstner, R. R. P. Singh, and A. P. Young, 71, 1629 (1993;
29. N. Elstner and A. P. Young, Phys. Rev. B 50, 6871 (1994).PRBMDO
30. J. Stephenson, J. Math. Phys. (N.Y.) 50, 1009 (1964); JMAPAQ
31. 11, 420 (1970).
32. A. Süto, Z. Phys. B 44, 121 (1981); ZPCMDN
33. Helv. Phys. Acta 54, 191 (1981); HPACAK
34. 54, 201 (1981).
35. G. D. Birkhoff, Ann. Math. 14, 42 (1912).ANMAAH
36. H. Whitney, Ann. Math. 33, 688 (1932); ANMAAH
37. Bull. Am. Math. Soc. 38, 572 (1932).BAMOAD
38. G. D. Birkhoff and D. C. Lewis, Trans. Am. Math. Soc. 60, 355 (1946).
39. B. Grünbaum and G. Shephard, Tilings and Patterns (Freeman, New York, 1987).
40. V. Matveev and R. Shrock, J. Phys. A 28, 5235 (1995).JPHAC5
41. F. Laves, Z. Kristallogr. 73, 202 (1930); ZEKRDZ
42. 78, 208 (1931).
43. K. I. Appel, W. Haken, and J. Koch, Illinois J. Math. 21, 429 (1977); K. I. Appel and W. Haken, ibid., 21, 491 (1977).
44. Our results for (Formula presented) are not difficult to obtain, although we have not been able to find them in the mathematical literature on graph theory, which deals mostly with proper (finite) graphs.
45. N. L. Biggs, Bull. London Math. Soc. 9, 54 (1977).
46. See, e.g., P. Lancaster and M. Tismenetsky, The Theory of Matrices, with Applications (Academic, New York, 1985); H. Minc, Nonnegative Matrices (Wiley, New York, 1988).
47. D. London, Duke Math. J. 33, 511 (1966).DUMJAO
48. The complete graph (Formula presented) is defined as the graph consisting of (Formula presented) vertices each of which is connected to all of the others with bonds.
49. The lower bound (Formula presented) for the reduced function (Formula presented) of Eq. (4.4) is not, in general, monotonically increasing over this range (Formula presented), although (Formula presented).
50. In passing, we note also that from (Formula presented) (Formula presented) and Eq. (4.12), it follows that (Formula presented) for (Formula presented).
51. J. F. Nagle, J. Combin. Theory 10, 42 (1971); see also G. A. Baker, Jr., ibid. 10, 217 (1971).
52. Although we have only listed our small-(Formula presented) series for (Formula presented) to (Formula presented) and this coincides with all terms of the corresponding expansion of our lower bound (Formula presented) a preliminary calculation (unpublished) of higher orders of the series for (Formula presented) indicates that this agreement continues for several more orders.
53. L. Onsager and M. Dupuis, in Rendiconti, Proceedings of the International School of Physics “Enrico Fermi,” Course 10, Bologna, 1960, edited by N. Zanichelli (Italian Physical Society, Bologna, 1960). E. H. Lieb and F. Y. Wu in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic, New York, 1972), Vol. 1, p. 331.
54. R. C. Read and G. F. Royle, Graph Theory, Combinatorics, and Applications (Wiley, New York, 1991), Vol. 2, p. 1009.