Spin-Orbital Locking, Emergent Pseudo-Spin, and Magnetic order in Na$_2$IrO$_3$

The nature of magnetic order in the honeycomb lattice Iridate Na$_2$IrO$_3$ is explored by considering trigonal crystal field effect and spin-orbit coupling. An effective Hamiltonian is derived in terms of an emergent pseudo-spin-1/2, resulting from a spin-orbital locking, which is different from $j_{\rm eff}=1/2$ that is obtained when the spin-orbit coupling dominates. The resulting Hamiltonian is anisotropic and frustrated. Mean field theory suggests a ground state with 4-sublattice {\em zig-zag} magnetic order in the relevant parameter regime, in conformity with experiments. Various properties of the phase, the spin-wave spectrum and experimental consequences are discussed. Our approach contrasts with the recent proposal of a Heisenberg-Kitaev system for this material, and we point out the intrinsic difficulties with the latter approach for describing the magnetic properties of Na$_2$IrO$_3$.