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Recently, realizations of Kitaev physics have been sought in the A2IrO3family of honeycomb iridates, originating from oxygen-mediated exchange throughedge-shared octahedra. However, for the j = 1/2 Mott insulator in thesematerials exchange from direct d-orbital overlap is relevant, and it wasproposed that a Heisenberg term should be added to the Kitaev model. Here weprovide the generic nearest-neighbour spin Hamiltonian when bothoxygen-mediated and direct overlap are present, containing a bond dependentoff-diagonal exchange in addition to Heisenberg and Kitaev terms. We analyzethis complete model using a combination of classical techniques and exactdiagonalization. Near the Kitaev limit, we find new magnetic phases, 120 degreeand incommensurate spiral order, as well as extended regions of zigzag andstripy order. Possible applications to Na2IrO3 and Li2IrO3 are discussed.

Generic spin model for the honeycomb iridates beyond the Kitaev limit