Hidden Order in Spin-Liquid Gd$_3$Ga$_5$O$_{12}$

Frustrated magnetic materials are promising candidates for new states of matter because lattice geometry suppresses conventional magnetic dipole order, potentially allowing non-dipole (``hidden") order to emerge in its place. However, an atomic-scale model of a hidden-order state has been difficult to obtain because microscopic probes are not directly sensitive to hidden order. We use a combination of neutron-scattering experiments and reverse Monte Carlo refinements to develop a model of the spin-liquid state in the canonical frustrated magnet Gd$_3$Ga$_5$O$_{12}$. We show that this state exhibits a hidden order which has three unusual properties. First, it is a collective phenomenon, in which multipoles are formed from ten-spin loops. Second, it is long-range, with a diverging correlation length. Third, it is a consequence of the interplay between antiferromagnetic spin correlations and local planar magnetic anisotropy, which allows it to be indirectly observed in our neutron-scattering experiments.