FIGURE 4. Possible spin fluctuations in the classical ground-state manifold.

a, Spin cluster surrounding a hexagon (shown in yellow) in the pyrochlore lattice of Fig. 1c. b, Generalized pattern of classical spin vectors on six neighbouring tetrahedra satisfying the condition that there be no net spin on any tetrahedron. c, Pattern of spin vectors satisfying the condition that = 0 for all tetrahedra. For such spin configurations, each tetrahedron has two pairs of antiparallel spins, and each hexagon has six collinear and antiferromagnetic spins as indicated by six red arrows in a. The energy of the spin cluster is independent of the orientation of the spin-loop directors.