Journal of Statistical Physics

, Volume 28, Issue 4, pp 621–632 | Cite as

The discrete Gaussian chain with 1/rn interactions: Exact results

  • K. H. Kjaer
  • H. J. Hilhorst


We show that the discrete Gaussian chain with interactionV(r) = 1/(r2−1/4) is self-dual. At the dual temperaturek B T = 1 we calculate the height-height correlation function and find that the system is rough. A duality relation is established for the temperature-dependent correlation function exponentη. We also consider interactionsV(r)−1/rn and show that absence of a phase transition for 2 <n < 3 implies absence of a phase transition for 1 <n < 2. All these results have their counterparts in a linear system of charges interacting through a potential which is asymptotically logarithmic (forn = 2) or power-law-like (forn ≠ 2.

Key words

Discrete Gaussian model long-range interactions self-duality 


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  1. 1.
    D. Ruelle,Statistical Mechanics: Rigorous Results (Benjamin, New York, 1969).Google Scholar
  2. 2.
    F. J. Dyson,Commun. Math. Phys. 12:91, 212 (1969).Google Scholar
  3. 3.
    P. W. Anderson, G. Yuval, and D. R. Hamann,Phys. Rev. B 1:4464 (1970).Google Scholar
  4. 4.
    P. W. Anderson and G. Yuval,Phys. Rev. B 1:1522 (1970).Google Scholar
  5. 5.
    B. Simon and A. D. Sokal,J. Stat. Phys. 25:679 (1981).Google Scholar
  6. 6.
    J. Fröhlich and T. Spencer,Phys. Rev. Lett. 46:1006 (1981).Google Scholar
  7. 7.
    J. Fröhlich and T. Spencer,Commun. Math. Phys. 81:527 (1981).Google Scholar
  8. 8.
    J. M. Kosterlitz and D. J. Thouless,J. Phys. C 6:1181 (1973).Google Scholar
  9. 9.
    S. T. Chui and J. D. Weeks,Phys. Rev. B 14:4978 (1976).Google Scholar
  10. 10.
    J. L. Cardy,J. Phys. A 14:1407 (1981).Google Scholar
  11. 11.
    A. J. F. Siegert,Physica 26:30 (1960).Google Scholar
  12. 12.
    J. Fröhlich,Commun. Math. Phys. 47:233 (1976).Google Scholar
  13. 13.
    R. H. Swendsen,Phys. Rev. B 18:492 (1978).Google Scholar
  14. 14.
    H. J. Hilhorst, H. N. J. Vogelij, C. van Leeuwen, and B. P. Th. Veltman, inNumerical Methods in the Study of Critical Phenomena, J. Della Dora, J. Demongeot, and B. Lacolle, eds., Springer Series in Energetics (Springer, Berlin, 1981).Google Scholar
  15. 15.
    P. M. Morse and H. F. Feshbach,Methods of Theoretical Physics (McGraw-Hill, New York, 1953). Chap. 4.8.Google Scholar

Copyright information

© Plenum Publishing Corporation 1982

Authors and Affiliations

  • K. H. Kjaer
    • 1
  • H. J. Hilhorst
    • 1
  1. 1.Laboratorium voor Technische NatuurkundeGA DelftThe Netherlands

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