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 μj=±1, with an interaction energy {Mathematical expression} where J(n) is positive and monotone decreasing, and the sums ΣJ(n) and Σ (log log n) [n3J(n)]-1 both converge. In particular, as conjectured by Kac and Thompson, a transition exists for J(n)=n-α when 1 < α < 2. A possible extension of these results to Heisenberg ferromagnets is discussed. © 1969 Springer-Verlag.