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, $\mathrm{I}c$.

Figure 1

Top: $[hhk]$ structure factor for $N=\infty$ [$\mathrm{\text{dark}}=\mathrm{\text{low}}$ intensity; the $x$ ($y$) axis corresponds to $h$ ($k$) ranging from $-2$ to 2]. Middle: Ising correlations at $T=0$ (bottom: Heisenberg, at $T/J=0.005$) from Monte Carlo simulations (symbols) compared to $T=0$ large-$N$ correlations (lines): $G_{11}^0(x)=⟨S_1(x)S_1(0)⟩$, for two inequivalent directions, [101] and [211], multiplied by the cube of the distance, $x^3$. Error bars are around $5\times {10}^{-6}{x}^3$ ($5\times {10}^{-5}{x}^3$). Different system sizes $L$ as indicated. There is no fitting parameter.

Figure 2

The centers of the tetrahedra define (the two sublattices of) the diamond lattice, denoted by black and gray dots. Vectors ${\widehat{e}}_{\kappa }$ define orientation of the diamond bonds, and the four sublattices of the pyrochlore lattice.