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

Abstract

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) [n ³ J(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.