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 $120^\circ$ magnetic order. The three Goldstone modes can be separated by point-group symmetry into a singlet $\alpha_{0}$ and a doublet $(\beta_x,\beta_y)$. The $\alpha_0$ singlet is described by the standard theory of a free relativistic scalar field. The field theory of the $(\beta_x,\beta_y)$ 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-neighbour 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 Mn$_3$X family with stacked kagome layers.