Correlations in Ising Ferromagnets. I

The following results are proved for a system of Ising spins σi = ±1 in zero magnetic field coupled by a purely ferromagnetic interaction of the form −Σij Jijσiσj with Jij ≥ 0, for arbitrary crystal lattice and range of interaction: (1) The binary correlation functions 〈σkσl〉 are always nonnegative (〈 〉 denotes a thermal average). (2) For arbitrary i, j, k, and l, 〈σiσjσk σl〉 ≥ 〈σiσj〉 〈σkσl〉. Consequences of these results, in particular the second, are: (i) 〈σkσl〉 never decreases if any Jij is increased. (ii) If an Ising model with ferromagnetic interactions exhibits a long-range order, this long-range order increases if additional ferromagnetic interactions are added. This last fact may be used to prove the existence of long-range order in a large class of two- and three-dimensional Ising lattices with purely ferromagnetic interactions of bounded or unbounded range.