, Volume 12, Issue 2, pp 91-107

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

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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.