Theory of Monomer-Dimer Systems

  • Ole J. Heilmann
  • Elliott H. Lieb
Chapter

Abstract

We investigate the general monomer-dimer partition function, P(x), which is a polynomial in the monomer activity, x, with coefficients depending on the dimer activities. Our main result is that P(x) has its zeros on the imaginary axis when the dimer activities are nonnegative. Therefore, no monomer-dimer system can have a phase transition as a function of monomer density except, possibly, when the monomer density is minimal (i.e. x = 0). Elaborating on this theme we prove the existence and analyticity of correlation functions (away from x = 0) in the thermodynamic limit. Among other things we obtain bounds on the compressibility and derive a new variable in which to make an expansion of the free energy that converges down to the minimal monomer density. We also relate the monomer-dimer problem to the Heisenberg and Ising models of a magnet and derive Christoffell-Darboux formulas for the monomer-dimer and Ising model partition functions. This casts the Ising model in a new light and provides an alternative proof of the Lee-Yang circle theorem. We also derive joint complex analyticity domains in the monomer and dimer activities. Our considerations are independent of geometry and hence are valid for any dimensionality.

Keywords

Entropy Methane Hexa Tungsten Hull 

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References

  1. 1.
    Roberts,J. K.: Proc. Roy. Soc. (London) A 152, 469 (1935).ADSGoogle Scholar
  2. 2.
    Proc. Roy. Soc. (London) A 161, 141 (1937).CrossRefGoogle Scholar
  3. 3.
    Proc. Cambridge Phil. Soc. 34, 399 (1938).CrossRefGoogle Scholar
  4. 4.
    Miller, A.R.: Proc. Cambridge Phil. Soc. 35, 293 (1939).Google Scholar
  5. 5.
    Readhead,P.A.: Trans. Faraday Soc. 57, 641 (1961).CrossRefGoogle Scholar
  6. 6.
    Rossington, D. R., Bost, R.: Surface Sci. 3, 202 (1965).ADSCrossRefGoogle Scholar
  7. 7.
    Lichtman, D., McQuistan, R. B.: J. Math. Phys. 8, 2441 (1967).ADSMATHCrossRefGoogle Scholar
  8. 8.
    McQuistan, R. B., Lichtman, D.: J. Math. Phys. 9, 1660 (1968).ADSCrossRefGoogle Scholar
  9. 9.
    J. Math. Phys. 10, 2205 (1969).Google Scholar
  10. 10.
    Lichtman,S.J.: J. Math. Phys. 11, 3095 (1970).MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Fowler, R. H., Rushbrooke, G. S.: Trans. Faraday Soc. 33, 1272 (1937).CrossRefGoogle Scholar
  12. 12.
    Guggenheim, A.: Trans. Faraday Soc. 33, 151 (1937).CrossRefGoogle Scholar
  13. 13.
    Chang,T.S.: Proc. Roy. Soc. (London) A 169, 512 (1939).ADSMATHCrossRefGoogle Scholar
  14. 14.
    Proc. Cambridge Phil. Soc. 35, 265 (1939).CrossRefGoogle Scholar
  15. 15.
    Miller, A. R.: Proc. Cambridge Phil. Soc. 38, 109 (1942).Google Scholar
  16. 16.
    Orr, W.J.C.: Trans. Faraday Soc. 40, 306 (1944).CrossRefGoogle Scholar
  17. 17.
    McGlashan, M. L.: Trans. Faraday Soc. 47, 1042 (1951).CrossRefGoogle Scholar
  18. 18.
    Miller, A. R.: Proc. Cambridge Phil. Soc. 39, 54 (1943).Google Scholar
  19. 19.
    Proc. Cambridge Phil. Soc. 39, 131 (1943).CrossRefGoogle Scholar
  20. 20.
    Orr, W.J. C.: Trans. Faraday Soc. 40, 320 (1944).CrossRefGoogle Scholar
  21. 21.
    Guggenheim, E. A.: Proc. Roy. Soc. (London) A 183, 203 (1944).ADSCrossRefGoogle Scholar
  22. 22.
    Proc. Roy. Soc. (London) A 183, 213 (1944).CrossRefGoogle Scholar
  23. 23.
    We shall not attempt to give a complete bibliography of the Flory-Huggins theory; the reader is referred to standard textbooks. The earliest references seem to be P. J. Flory: J. Chem. Phys. 10, 51 (1942) and Huggins: Ann. N. Y. Acad. Sci. 43, 9 (1942).Google Scholar
  24. 24.
    Guggenheim, E. A.: Mixtures, Chapter X. Oxford: Claredon Press 1952.Google Scholar
  25. 25.
    Rushbrooke, G. S., Scoins, H. I., Wakefield, A. J.: Discussions Farad. Soc. 15, 57 (1953).Google Scholar
  26. 26.
    Travena, D.H.: Proc. Phys. Soc. 84, 969 (1964).ADSCrossRefGoogle Scholar
  27. 27.
    Nagle,J. F.: Phys. Rev. 152, 190 (1966).ADSCrossRefGoogle Scholar
  28. 28.
    Gaunt, D. S.: Phys. Rev. 179, 174 (1969).ADSCrossRefGoogle Scholar
  29. 29.
    Bellemans, A., Fuks, S.: Physica 50, 348 (1970).ADSCrossRefGoogle Scholar
  30. 30.
    Runnels, L. K.: J. Math. Phys. 11, 842 (1970).ADSCrossRefGoogle Scholar
  31. 31.
    Baxter, R. J.: J. Math. Phys. 9, 650 (1968).ADSCrossRefGoogle Scholar
  32. 32.
    Craen,J.van, Bellemans, A.: Bull. Acad. Pol. Sci. 19, 45 (1971).Google Scholar
  33. 33.
    Hammersley,J. M.: In: Proceedings of the 2nd Annual Conference on Computational Physics, pp. 1–8 (Institute of Physics and Physical Society, London (1970)).Google Scholar
  34. 34.
    Baxendale, J. H., Enüstün, B. V., Stern, J.: Phil. Trans. Roy. Soc. (London) A 243, 169 (1951).ADSCrossRefGoogle Scholar
  35. 35.
    Everett, D. H., Penney, M. F.: Proc. Roy. Soc. (London) A 212, 164 (1952).ADSCrossRefGoogle Scholar
  36. 36.
    Tompa, H.: J. Chem. Phys. 16, 292 (1948).ADSCrossRefGoogle Scholar
  37. 37.
    Brondsted,J. N., Koefoed,J.: Kgl. Danske Videnskab. Selskob. Mat-Fys. Medd. 22, No. 17 (1946).Google Scholar
  38. 38.
    Tompa, H.: Trans. Faraday Soc. 45, 101 (1949).CrossRefGoogle Scholar
  39. 39.
    Pizzini,S., Morlotti, R., Wagner, V.: J. Electrochem. Soc. 116, 915 (1969).CrossRefGoogle Scholar
  40. 40.
    Cohen, E. G. D., De Boer, J., Salsburg, Z. W.: Physica 21, 137 (1955).ADSMATHCrossRefGoogle Scholar
  41. 41.
    Conway, B. E., Verall, R. E.: J. Phys. Chem. 70, 1473 (1966).CrossRefGoogle Scholar
  42. 42.
    Fisher, M. E., Temperley, H. N. V.: Rev. Mod. Phys. 32, 1029 (1960).ADSCrossRefGoogle Scholar
  43. 43.
    Katsura,S., Inawashiro,S.: Rev. Mod. Phys. 32, 1031 (1960).ADSCrossRefGoogle Scholar
  44. 44.
    Kasteleyn,P. W.: Physica, Gray. 27, 1209 (1961).ADSMATHCrossRefGoogle Scholar
  45. 45.
    Temperley, H. N. V., Fisher, M. E.: Phil. Mag. Serie 8 6, 1061 (1961).MathSciNetGoogle Scholar
  46. 46.
    Fisher, M. E.: Phys. Rev. 124, 1664 (1961).MathSciNetADSMATHCrossRefGoogle Scholar
  47. 47.
    Kasteleyn, P. W.: J. Math. Phys. 4, 287 (1963).MathSciNetADSCrossRefGoogle Scholar
  48. 48.
    Montroll, E. W.: In: Applied combinatorial mathematics (Ed. F. Beckenbach). New York: J. Wiley & Sons, 1964.Google Scholar
  49. 49.
    Lieb, E. H.: J. Math. Phys. 8, 2339 (1967).ADSCrossRefGoogle Scholar
  50. 50.
    Gibberd, R. W.: Can. J. Phys. 46, 1681 (1968).ADSCrossRefGoogle Scholar
  51. 51.
    Wu,T.T.: J. Math. Phys. 3, 1265 (1962).ADSMATHCrossRefGoogle Scholar
  52. 52.
    Ferdinand, A.E.: J. Math. Phys. 8, 2332 (1967).ADSCrossRefGoogle Scholar
  53. 53.
    Hammersley, J. M., Feuerverger, A., Izenman, A., Mahani, S.: J. Math. Phys. 10, 443 (1969).ADSMATHCrossRefGoogle Scholar
  54. 54.
    Fisher, M. E., Stephenson, J.: Phys. Rev. 132, 1411 (1963).MathSciNetADSMATHCrossRefGoogle Scholar
  55. 55.
    Hartwig, R. E.: J. Math. Phys. 7, 286 (1966).MathSciNetADSCrossRefGoogle Scholar
  56. 56.
    Bondy, J. A., Welsh, D. J. A.: Proc. Cambridge Phil. Soc. Math. Phys. Sci. 62, 503 (1966).MathSciNetGoogle Scholar
  57. 57.
    Hammersley,J. M.: Proc. Cambridge Phil. Soc. Math. Phys. Sci. 64, 455 (1968).MathSciNetMATHGoogle Scholar
  58. 58.
    Menon, V. V.: J. Inst. Math. Appl. 6, 341 (1970).MathSciNetMATHCrossRefGoogle Scholar
  59. 59.
    In: Research papers in statistics. Festschrift für J. Neyman, p. 125 (Editor, F. N. David ). New York: John Wiley & Sons 1966.Google Scholar
  60. 60.
    Heilmann, O. J.: Existence of phase transitions in certain lattice gases with repulsive potentials (to be published).Google Scholar
  61. 61.
    Lieb, E. H.: Phys. Rev. Letters 24, 1412 (1970).ADSCrossRefGoogle Scholar
  62. 62.
    Kunz, H.: Phys. Letters 32A, 311 (1970).ADSCrossRefGoogle Scholar
  63. 63.
    Gruber, C., Kunz, H.: Commun. math. Phys. 22, 133 (1971).MathSciNetADSCrossRefGoogle Scholar
  64. 64.
    Dobrushin, R. L.: Funct. Anal. Appl. 2, No. 4, 44 (1968), (English translation 2, 302 (1968)).Google Scholar
  65. 65.
    Essam, J. W., Fisher, M. E.: Rev. Mod. Phys. 42, 271 (1970).MathSciNetADSCrossRefGoogle Scholar
  66. 66.
    Szegö, G.: Orthogonal polynomials (American Mathematical Society, Colloquium Publications Vol. XXIII, Providence 1939 ).Google Scholar
  67. 67.
    Ruelle, D.: Statistical mechanics. New York: W. A. Benjamin 1969.Google Scholar
  68. 68.
    Fisher, M. E.: J. Math. Phys. 7, 1776 (1966).ADSCrossRefGoogle Scholar
  69. 69.
    Lee, T. D., Yang, C. N.: Phys. Rev. 87, 410 (1952).MathSciNetADSMATHCrossRefGoogle Scholar
  70. 70.
    Asano,T.: J. Phys. Soc. J.pan 29, 350 (1970); Phys. Rev. Letters 24, 1409 (1970).Google Scholar
  71. 71.
    Suzuki, M., Fisher, M. E.: J. Math. Phys. 12, 235 (1971).MathSciNetADSCrossRefGoogle Scholar
  72. 72.
    Ginibre,J.: Phys. Letters 24A, 223 (1967).ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1972

Authors and Affiliations

  • Ole J. Heilmann
    • 2
  • Elliott H. Lieb
    • 1
    • 2
  1. 1.Department of Mathematics2-375 Massachusetts Institute of TechnologyCambridgeUSA
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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