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$\mathrm{Mn}_3\mathrm{X}$ compounds in which the magnetic $\mathrm{Mn}$ atomsform AB-stacked kagome lattices have received a tremendous amount of attentionsince the observation of the anomalous Hall effect in$\mathrm{Mn}_3\mathrm{Ge}$ and $\mathrm{Mn}_3\mathrm{Sn}$. Although themagnetic ground state has been known for some time to be an inverse triangularstructure with an induced in-plane magnetic moment, there have been severalcontroversies about the minimal magnetic Hamiltonian. We present a generalsymmetry-based model for these compounds that includes a previously unreportedinterplane Dzyaloshinskii-Moriya interaction, as well as anisotropic exchangeinteractions. The latter are shown to compete with the single-ion anisotropywhich strongly affects the ground state configurations and elementary spin-waveexcitations. Finally, we present the calculated elastic and inelastic neutronscattering intensities and point to experimental assessment of the types ofmagnetic anisotropy in these compounds that may be important.

physical review. b./physical review. b

Anisotropic magnetic interactions in hexagonal AB-stacked kagome lattice  structures: Applications to $\mathrm{Mn}_3\mathrm{X}$ ($\mathrm{X}$ =  $\mathrm{Ge}$, $\mathrm{Sn}$, $\mathrm{Ga}$) compounds