Consider the classical 2D-model of a Heisenberg easy-plane AFM, with the Hamiltonian
Here J > 0 is the exchange constant, describes anisotropy with the xy-plane as the easy-plane. Spins are classical vectors on a 2D-square lattice with lattice constant a. denotes lattice sites of one sublattice, and 's are the set of displacements to the nearest-neighbors on the other sublattice. We are interested in the small anisotropy case .
A continuum model of AFM's can be derived from (1) in the usual way, see ,,. We define the magnetization vector and the sublattice magnetization vector on the set of nearest-neighbor pairs, with the constraints , . Then, for low frequencies and small gradients, , the magnetization of an AFM is small, . The magnetization can be considered as a ``slave'' variable, and can be expressed as , where is a susceptibility defined below [Eq. (3)]. After eliminating one obtains equations for only. Using the usual angular variables, () these equations can be written in the form,
where c is the magnon phase velocity, is the characteristic length scale. For the Hamiltonian (1)