  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 $\beta$-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.

Strongly Correlated Electrons