  Recently, realizations of Kitaev physics have been sought in the A2IrO3
family of honeycomb iridates, originating from oxygen-mediated exchange through
edge-shared octahedra. However, for the j = 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 degree
and incommensurate spiral order, as well as extended regions of zigzag and
stripy order. Possible applications to Na2IrO3 and Li2IrO3 are discussed.

Strongly Correlated Electrons