Correlations in Ising Ferromagnets. I

TLDR

The authors aim to use correlation inequalities to prove long‑range order in a broad class of two‑ and three‑dimensional Ising lattices with purely ferromagnetic interactions. They establish that all binary spin correlations are nonnegative and that higher‑order correlations satisfy a factorization inequality, implying monotonicity of correlations with respect to interaction strengths. These results show that correlations never decrease when any Jij is increased and that adding ferromagnetic interactions strengthens existing long‑range order.

Abstract

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 −Σi<j 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.