An Exact Criterion for Finite-Temperature Order in One-Dimensional Ising Ferromagnets

Which long-range interactions can order an Ising chain? We give an exact criterion for every summable, translation-invariant ferromagnetic coupling. Order at some nonzero temperature exists if and only if reversing a block of N spins costs at least a positive multiple of log N for all sufficiently large N. We prove sufficiency without monotonicity or regular decay, including sparse couplings; necessity follows from Berbee's theorem. The criterion extends to finite-state Potts chains, and finite-range susceptibilities give monotone bounds converging to the critical temperature.