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The recently discovered three-dimensional hyperhoneycomb iridate,$\beta$-Li$_2$IrO$_3$, has raised hopes for the realization of dominant Kitaevinteraction between spin-orbit entangled local moments due to its near-ideallattice structure. If true, this material may lie close to the sought-afterquantum spin liquid phase in three dimensions. Utilizing ab-initio electronicstructure calculations, we first show that the spin-orbit entangled basis,$j_{\rm eff}$=1/2, correctly captures the low energy electronic structure. Theeffective spin model derived in the strong coupling limit supplemented by theab-initio results is shown to be dominated by the Kitaev interaction. Wedemonstrated that the possible range of parameters is consistent with anon-coplanar spiral magnetic order found in a recent experiment. All of theseanalyses suggest that $\beta$-Li$_2$IrO$_3$ may be the closest among knownmaterials to the Kitaev spin liquid regime.

Predominance of the Kitaev interaction in a three-dimensional honeycomb iridate: from ab-initio to spin model