Communications in Mathematical Physics

, Volume 25, Issue 3, pp 190–232 | Cite as

Theory of monomer-dimer systems

  • Ole J. Heilmann
  • Elliott H. Lieb
Article

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 thatP(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 fromx=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.

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References

  1. 1.
    Roberts, J.K.: Proc. Roy. Soc. (London) A152, 469 (1935).Google Scholar
  2. 2.
    —— Proc. Roy. Soc. (London) A161, 141 (1937).Google Scholar
  3. 3.
    —— Proc. Cambridge Phil. Soc.34, 399 (1938).Google 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).Google Scholar
  6. 6.
    Rossington, D. R., Bost, R.: Surface Sci.3, 202 (1965).Google Scholar
  7. 7.
    Lichtman, D., McQuistan, R. B.: J. Math. Phys.8, 2441 (1967).Google Scholar
  8. 8.
    McQuistan, R. B., Lichtman, D.: J. Math. Phys.9, 1660 (1968).Google Scholar
  9. 9.
    —— J. Math. Phys.10, 2205 (1969).Google Scholar
  10. 10.
    —— Lichtman, S. J.: J. Math. Phys.11, 3095 (1970).Google Scholar
  11. 11.
    Fowler, R. H., Rushbrooke, G. S.: Trans. Faraday Soc.33, 1272 (1937).Google Scholar
  12. 12.
    Guggenheim, A.: Trans. Faraday Soc.33, 151 (1937).Google Scholar
  13. 13.
    Chang, T. S.: Proc. Roy. Soc. (London) A169, 512 (1939).Google Scholar
  14. 14.
    —— Proc. Cambridge Phil. Soc.35, 265 (1939).Google 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).Google Scholar
  17. 17.
    McGlashan, M. L.: Trans. Faraday Soc.47, 1042 (1951).Google Scholar
  18. 18.
    Miller, A. R.: Proc. Cambridge Phil. Soc.39, 54 (1943).Google Scholar
  19. 19.
    —— Proc. Cambridge Phil. Soc.39, 131 (1943).Google Scholar
  20. 20.
    Orr, W. J. C.: Trans. Faraday Soc.40, 320 (1944).Google Scholar
  21. 21.
    Guggenheim, E. A.: Proc. Roy. Soc. (London) A183, 203 (1944).Google Scholar
  22. 22.
    —— Proc. Roy. Soc. (London) A183, 213 (1944).Google 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).Google Scholar
  27. 27.
    Nagle, J. F.: Phys. Rev.152, 190 (1966).Google Scholar
  28. 28.
    Gaunt, D. S.: Phys. Rev.179, 174 (1969).Google Scholar
  29. 29.
    Bellemans, A., Fuks, S.: Physica50, 348 (1970).Google Scholar
  30. 30.
    Runnels, L. K.: J. Math. Phys.11, 842 (1970).Google Scholar
  31. 31.
    Baxter, R. J.: J. Math. Phys.9, 650 (1968).Google 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) A243, 169 (1951).Google Scholar
  35. 35.
    Everett, D. H., Penney, M. F.: Proc. Roy. Soc. (London) A212, 164 (1952).Google Scholar
  36. 36.
    Tompa, H.: J. Chem. Phys.16, 292 (1948).Google Scholar
  37. 37.
    Brøndsted, 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).Google Scholar
  39. 39.
    Pizzini, S., Morlotti, R., Wagner, V.: J. Electrochem. Soc.116, 915 (1969).Google Scholar
  40. 40.
    Cohen, E. G. D., De Boer, J., Salsburg, Z. W.: Physica21, 137 (1955).Google Scholar
  41. 41.
    Conway, B. E., Verall, R. E.: J. Phys. Chem.70, 1473 (1966).Google Scholar
  42. 42.
    Fisher, M. E., Temperley, H. N. V.: Rev. Mod. Phys.32, 1029 (1960).Google Scholar
  43. 43.
    Katsura, S., Inawashiro, S.: Rev. Mod. Phys.32, 1031 (1960).Google Scholar
  44. 44.
    Kasteleyn, P. W.: Physica, Grav.27, 1209 (1961).Google Scholar
  45. 45.
    Temperley, H. N. V., Fisher, M. E.: Phil. Mag. Serie 86, 1061 (1961).Google Scholar
  46. 46.
    Fisher, M. E.: Phys. Rev.124, 1664 (1961).Google Scholar
  47. 47.
    Kasteleyn, P. W.: J. Math. Phys.4, 287 (1963).Google 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).Google Scholar
  50. 50.
    Gibberd, R. W.: Can. J. Phys.46, 1681 (1968).Google Scholar
  51. 51.
    Wu, T. T.: J. Math. Phys.3, 1265 (1962).Google Scholar
  52. 52.
    Ferdinand, A. E.: J. Math. Phys.8, 2332 (1967).Google Scholar
  53. 53.
    Hammersley, J. M., Feuerverger, A., Izenman, A., Mahani, S.: J. Math. Phys.10, 443 (1969).Google Scholar
  54. 54.
    Fisher, M. E., Stephenson, J.: Phys. Rev.132, 1411 (1963).Google Scholar
  55. 55.
    Hartwig, R. E.: J. Math. Phys.7, 286 (1966).Google Scholar
  56. 56.
    Bondy, J. A., Welsh, D. J. A.: Proc. Cambridge Phil. Soc. Math. Phys. Sci.62, 503 (1966).Google Scholar
  57. 57.
    Hammersley, J. M.: Proc. Cambridge Phil. Soc. Math. Phys. Sci.64, 455 (1968).Google Scholar
  58. 58.
    —— Menon, V. V.: J. Inst. Math. Appl.6, 341 (1970).Google 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. Letters24, 1412 (1970).Google Scholar
  62. 62.
    Kunz, H.: Phys. Letters32, 311 (1970).Google Scholar
  63. 63.
    Gruber, C., Kunz, H.: Commun. math. Phys.22, 133 (1971).Google Scholar
  64. 64.
    Dobrushin, R. L.: Funct. Anal. Appl.2, No. 4, 44 (1968), (English translation2, 302 (1968)).Google Scholar
  65. 65.
    Essam, J. W., Fisher, M. E.: Rev. Mod. Phys.42, 271 (1970).Google 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).Google Scholar
  69. 69.
    Lee, T. D., Yang, C. N.: Phys. Rev.87, 410 (1952).Google Scholar
  70. 70.
    Asano, T.: J. Phys. Soc. Japan29, 350 (1970); Phys. Rev. Letters24, 1409 (1970).Google Scholar
  71. 71.
    Suzuki, M., Fisher, M. E.: J. Math. Phys.12, 235 (1971).Google Scholar
  72. 72.
    Ginibre, J.: Phys. Letters24, 223 (1967).Google Scholar

Copyright information

© Springer-Verlag 1972

Authors and Affiliations

  • Ole J. Heilmann
    • 1
  • Elliott H. Lieb
    • 1
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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