Spin waves and stability of zigzag order in the Hubbard model with spin-dependent hopping terms - Application to the honeycomb lattice compounds $\mathrm{N}{\mathrm{a}}_2\mathrm{I}\mathrm{r}{\mathrm{O}}_3$ and $\alpha -\mathrm{R}\mathrm{u}\mathrm{C}{\mathrm{l}}_3$

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 $\alpha -\mathrm{R}\mathrm{u}\mathrm{C}{\mathrm{l}}_3$ and $\mathrm{N}{\mathrm{a}}_2\mathrm{I}\mathrm{r}{\mathrm{O}}_3$.