Theory of spin waves in a hexagonal antiferromagnet

We construct a field-theoretic description of spin waves in hexagonal antiferromagnets with three magnetic sublattices and coplanar ${120}^{\ensuremath{\circ}}$ magnetic order. The three Goldstone modes can be separated by point-group symmetry into a singlet ${\ensuremath{\alpha}}₀$ and a doublet $\mathbit{\ensuremath{\beta}}$. The ${\ensuremath{\alpha}}₀$ singlet is described by the standard theory of a free relativistic scalar field. The field theory of the $\mathbit{\ensuremath{\beta}}$ doublet is analogous to the theory of elasticity of a two-dimensional isotropic solid with distinct longitudinal and transverse ``speeds of sound.'' The well-known Heisenberg models on the triangular and kagome lattices with nearest-neighbor exchange turn out to be special cases with accidental degeneracy of the spin-wave velocities. The speeds of sound can be readily calculated for any lattice model. We apply this approach to the compounds of the ${\mathrm{Mn}}₃X$ family with stacked kagome layers.