Long range order for lattice dipoles

TL;DR

This paper proves that classical Heisenberg spins with dipole interactions on cubic lattices exhibit long-range orientational order at low temperatures, confirming a long-standing conjecture using reflection positivity and spin wave analysis.

Contribution

It provides a rigorous proof of long-range order in dipolar Heisenberg models and verifies a conjecture on spin wave dispersion relations.

Findings

Long-range orientational order established at low temperatures.

Validation of the spin wave dispersion conjecture by Froehlich and Spencer.

Application of reflection positivity methods to dipolar spin systems.

Abstract

We consider a system of classical Heisenberg spins on a cubic lattice in dimensions three or more, interacting via the dipole-dipole interaction. We prove that at low enough temperature the system displays orientational long range order, as expected by spin wave theory. The proof is based on reflection positivity methods. In particular, we demonstrate a previously unproven conjecture on the dispersion relation of the spin waves, first proposed by Froehlich and Spencer, which allows one to apply infrared bounds for estimating the long distance behavior of the spin-spin correlation functions.