The Dzyaloshinskii-Moriya Anisotropy in Hyper-Kagome Lattice for $\mbox{Na}_4 \mbox{Ir}_3 \mbox{O}_8 $

The effect of quantum correction on the classical spin order in the iridium antiferromagnet $\mbox{Na}_4 \mbox{Ir}_3 \mbox{O}_8 $ are considered. The spin-wave spectrum is calculated to the linear order. The nearest-neighbor exchange interactions between the $\mbox{Ir}$spins as well as the Dzyaloshinskii-Moriya interactions are taken into account. Numerical simulation by exact diagonalization and experimental results are used to compare with the theoretical prediction.