Communications in Mathematical Physics

, Volume 12, Issue 2, pp 91–107 | Cite as

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

  • Freeman J. Dyson


Existence of a phase-transition is proved for an infinite linear chain of spins μ j =±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.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Interaction Energy 
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Copyright information

© Springer-Verlag 1969

Authors and Affiliations

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

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