Classical antiferromagnet on a hyper-kagome lattice

Motivated by recent experiments on Na$_4$Ir$_3$O$_8$ [Y. Okamoto {\it{et al.}}, Phys. Rev. Lett. 99, 167402 (2007)], we study the classical antiferromagnet on a frustrated three-dimensional lattice obtained by selectively removing one of four sites in each tetrahedron of the pyrochlore lattice. This ``hyper-kagome'' lattice consists of corner-sharing triangles. We present (J. Hopkinson {\it{et al.}}, Phys. Rev. Lett. 99, 037201 (2007)) the results of large-$N$ mean field theory and Monte Carlo computations on $O(N)$ classical spin models. We find the classical ground states to be highly degenerate. Nonetheless, at low temperatures, nematic order emerges via ``order by disorder'' in the Heisenberg model ($N$=3), representing the dominance of coplanar spin configurations. Above this transition, the spin-spin correlations show a dipolar form which can be understood to arise from a generalized ``Gauss' law'' constraint. The relevance of these results to ongoing neutron scattering measurements will be discussed.