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Topological and magnetic phases with strong spin-orbit coupling on the hyperhoneycomb lattice

We study the general phase diagram of correlated electrons for iridium-based (Ir) compounds on the hyperhoneycomb lattice---a crystal structure where the Ir$^{4+}$ ions form a three-dimensional network with three-fold coordination recently realized in the $β$-Li${}_{2}$IrO${}_{3}$ compound. Using a combination of microscopic derivations, symmetry analysis, and density functional calculations, we determine the general model for the electrons occupying the $j_{\text{eff}}=1/2$ orbitals at the Ir$^{4+}$ sites. In the non-interacting limit, we find that this model allows for both topological and trivial electronic band insulators along with metallic states. The effect of Hubbard-type electron-electron repulsion on the above electronic structure in stabilizing $\mathbf{q}=\mathbf{0}$ magnetic order reveals a phase diagram with continuous phase transition between a topological band insulator and a Neel ordered magnetic insulator.