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Novel magnetic ordering on the honeycomb lattice due to emergent weakanisotropic interactions generated by the mixing between the $J=1/2$ sector andthe magnetically inactive 3/2 sector is investigated in a three-orbitalinteracting electron model in the absence of Hund's coupling. Self-consistentdetermination of magnetic order yields anisotropic N\'{e}el and zigzag ordersfor different parameter regimes, highlighting the effect of the emergentsingle-ion anisotropy. Study of magnon excitations shows extremely small magnonenergy scale compared to the hopping energy scale, and enhancement ofanisotropy effects for smaller spin-orbit coupling. These results account forseveral features of the honeycomb lattice compounds such as $\rm Na_2 Ir O_3$and $\rm Ru Cl_3$, where the leading order anisotropic interactions within themagnetically active $J=1/2$ sector are completely quenched due to theedge-sharing octahedra.

Magnetic order and anisotropic interactions induced by mixing between  the $J=1/2$ and $3/2$ sectors in spin-orbit coupled honeycomb-lattice  compounds