Generic spin model for the honeycomb iridates beyond the Kitaev limit

Recently, realizations of Kitaev physics have been sought in the A$_2$IrO$_3$ family of honeycomb iridates, originating from oxygen-mediated exchange through edge-shared octahedra. However, for the $j_{\rm eff} = 1/2$ Mott insulator in these materials exchange from direct $d$-orbital overlap is relevant, and it was proposed that a Heisenberg term should be added to the Kitaev model. Here we provide the generic nearest-neighbour spin Hamiltonian when both oxygen-mediated and direct overlap are present, containing a bond dependent off-diagonal exchange in addition to Heisenberg and Kitaev terms. We analyze this complete model using a combination of classical techniques and exact diagonalization. Near the Kitaev limit we find new magnetic phases: 120${}^\circ$ and incommensurate spiral order, as well as extended regions of zigzag and stripy order. Possible applications to Na$_2$IrO$_3$ and Li$_2$IrO$_3$ are discussed.