Dipolar Spin Correlations in Classical Pyrochlore Magnets

We study spin correlations for the highly frustrated classical pyrochlore lattice antiferromagnets with $O(N)$ symmetry in the limit $T\to 0$. We conjecture that a local constraint obeyed by the extensively degenerate ground states dictates a dipolar form for the asymptotic spin correlations, at all $N\ne 2$ for which the system is paramagnetic down to $T=0$. We verify this conjecture in the cases $N=1$ and $N=3$ by simulations and to all orders in the $1/N$ expansion about the solvable $N=\infty$ limit. Remarkably, the $N=\infty$ formulas are an excellent fit, at all distances, to the correlators at $N=3$ and even at $N=1$. Thus we obtain a simple analytical expression also for the correlations of the equivalent models of spin ice and cubic water ice, $\mathrm{I}c$.