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We consider spin-orbit coupling effects in Na$_4$Ir$_3$O$_8$, a material inwhich Ir$^{4+}$ spins form an hyper-kagom\'{e} lattice, a three-dimensionalnetwork of corner-sharing triangles. We argue that both low temperaturethermodynamic measurements and the impurity susceptibility induced by dilutesubstitution of Ti for Ir are suggestive of significant spin-orbit effects.Because of uncertainties in the crystal-field parameters, we consider twolimits in which the spin-orbit coupling is either weak or strong compared tothe non-cubic atomic splittings. A semi-microscopic calculation of the exchangeHamiltonian confirms that indeed large antisymmetric Dzyaloshinskii-Moriya (DM)and/or symmetric exchange anisotropy may be present. In the strong spin-orbitlimit, the Ir-O-Ir superexchange contribution consists of unfrustrated strongsymmetric exchange anisotropy, and we suggest that spin-liquid behavior isunlikely. In the weak spin-orbit limit, and for strong spin-orbit and directIr-Ir exchange, the Hamiltonian consists of Heisenberg and DM interactions. TheDM coupling is parametrized by a three component DM vector (which must bedetermined empirically). For a range of orientation of this vector, frustrationis relieved and an ordered state occurs. For other orientations, even theclassical ground states are very complex. We perform spin-wave and exactdiagonalization calculations which suggest the persistence of a quantum spinliquid in the latter regime. Applications to Na$_4$Ir$_3$O$_8$ and broaderimplications are discussed.

Spin-orbit effects in Na$_4$Ir$_3$O$_8$, a hyper-kagom\'{e} lattice antiferromagnet