Spin-orbit effects in Na₄Ir₃O₈: A hyper-kagome lattice antiferromagnet

We consider spin-orbit coupling effects in Na₄Ir₃O₈, a material in which Ir⁴⁺ spins form an hyper-kagome lattice, a three-dimensional network of corner-sharing triangles. We argue that both low-temperature thermodynamic measurements and the impurity susceptibility induced by dilute substitution of Ti for Ir are suggestive of significant spin-orbit effects. Because of uncertainties in the crystal-field parameters, we consider two limits in which the spin-orbit coupling is either weak or strong compared to the noncubic atomic splittings. A semi-microscopic calculation of the exchange Hamiltonian confirms that indeed large antisymmetric Dzyaloshinskii-Moriya (DM) and/or symmetric exchange anisotropy may be present. In the strong spin-orbit limit, the Ir-O-Ir superexchange contribution consists of unfrustrated strong symmetric exchange anisotropy, and we suggest that spin-liquid behavior is unlikely. In the weak spin-orbit limit, and for strong spin-orbit and direct Ir-Ir exchange, the Hamiltonian consists of Heisenberg and DM interactions. The DM coupling is parametrized by a three-component DM vector (which must be determined empirically). For a range of orientation of this vector, frustration is relieved and an ordered state occurs. For other orientations, even the classical ground states are very complex. We perform spin-wave and exact diagonalization calculations, which suggest the persistence of a quantum spin liquid in the latter regime. Applications to Na₄Ir₃O₈ and broader implications are discussed.