Communications in Mathematical Physics

, Volume 12, Issue 2, pp 91–107

Existence of a phase-transition in a one-dimensional Ising ferromagnet

  • Freeman J. Dyson
Article

Abstract

Existence of a phase-transition is proved for an infinite linear chain of spins μj=±1, with an interaction energy
$$H = - \sum J(i - j)\mu _i \mu _j ,$$
whereJ(n) is positive and monotone decreasing, and the sums ΣJ(n) and Σ (log logn) [n3J(n)]−1 both converge. In particular, as conjectured byKac andThompson, a transition exists forJ(n)=n−α when 1 < α < 2. A possible extension of these results to Heisenberg ferromagnets is discussed.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Gallavotti, G., andS. Miracle-Sole: Commun. Math. Phys.5, 317 (1967).Google Scholar
  2. 2.
    Rushbrooke, G., andH. Ursell: Proc. Cambridge Phil. Soc.44, 263 (1948).Google Scholar
  3. 3.
    Baur, M., andL. Nosanow: J. Chem. Phys.37, 153 (1962).Google Scholar
  4. 4.
    Ruelle, D.: Commun. Math. Phys.9, 267 (1968).Google Scholar
  5. 5.
    Kac, M., andC. J. Thompson: Critical behavior of several lattice models with long-range interaction. Preprint, Rockefeller University, 1968.Google Scholar
  6. 6.
    Thompson, C. J.: (private communication) now believes that there is no transition forJ(n)=n −2.Google Scholar
  7. 7.
    Griffiths, R. B.: Commun. Math. Phys.6, 121 (1967).Google Scholar
  8. 8.
    —— J. Math. Phys.8, 478 (1967). See alsoKelly, D. G., andS. Sherman: J. Math. Phys.9, 466 (1968).Google Scholar
  9. 9.
    Hardy, G. H., J. E. Littlewood, andG. Polya: Inequalities, p. 43. Cambridge Univ. Press 1934.Google Scholar
  10. 10.
    Gallavotti, G., S. Miracle-Sole, andD. W. Robinson: Phys. Letters25 A, 493 (1967).Google Scholar

Copyright information

© Springer-Verlag 1969

Authors and Affiliations

  • Freeman J. Dyson
    • 1
  1. 1.Institute for Advanced StudyPrinceton

Personalised recommendations