Generalizations of the Kitaev-Heisenberg model and connections to experiment

Generalizations of Kitaev-Heisenberg spin models are both interesting on their own right and potentially relevant for the Mott insulating layered Iridates A$_2$IrO$_3$ (A=Na,Li) and other complex oxides of 5d transition metal compounds. To describe Na$_2$IrO$_3$ and Li$_2$IrO$_3$ we propose the Kitaev-Heisenberg-$J_2$-$J_3$ model, a combination of the Kitaev honeycomb model and the Heisenberg model with all three nearest neighbor couplings $J_1$, $J_2$ and $J_3$. A rich phase diagram is obtained at the classical level, including the experimentally suggested \textit{zigzag} ordered phase; as well as the \textit{stripy} phase, which extends from the Kitaev-Heisenberg limit to the $J_1$-$J_2$-$J_3$ one. Combining the experimentally observed spin order with the optimal fitting to the uniform magnetic susceptibility data gives an estimate of possible parameter values, which in turn reaffirms the necessity of including both the Kitaev and farther neighbor couplings in describing the materials. Generalizations for other spin-orbit coupled Mott insulating 5d transition metal oxides with Kitaev-Heisenberg type models exhibit related magnetically ordered phases as well as different spin liquid phases.