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Spin waves in the zigzag ordered state on a honeycomb lattice areinvestigated within a Hubbard model with spin-dependent hopping terms. Roles ofthe emergent Kitaev, Heisenberg, Dzyaloshinskii-Moriya, andsymmetric-off-diagonal spin interactions are investigated on the stability ofthe zigzag order. While the DM interactions are found to destabilize the zigzagorder, the secondary spin-dependent hopping terms (associated with structuraldistortions) are shown to strongly stabilize the zigzag order and account formagnetocrystalline anisotropy, easy axis, and spin wave gap. The calculatedspin wave dispersion and energy scale are in good agreement with inelasticneutron scattering measurements on $\rm \alpha - RuCl_3$ and $\rm Na_2 Ir O_3$.

journal of magnetism and magnetic materials

Spin waves and stability of zigzag order in the Hubbard model with  spin-dependent hopping terms - Application to the honeycomb lattice compounds  ${\rm Na_2 Ir O_3}$ and ${\rm \alpha - Ru Cl_3}$