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