Normal modes in an anisotropic antiferromagnet

The bosonic equivalent Hamiltonian of a Heisenberg antiferromagnet with anisotropy is not diagonalizable in a simple way when one takes into account the bilinear contributions in creation and destruction operators coming from the normal ordering of interaction terms, even if the kinematical and the dynamical interactions are neglected. We give the solution of this problem in implicit form for arbitrary dimensionality and lattice with center of symmetry. For all kinds of anisotropy we have considered (single-ion, exchange, mean-field) we obtain the following compact expression for the spin-wave spectrum: $E_k=E_{\mathrm{ZB}}{\{1-[1-{\left(\frac{E_0}{E_{\mathrm{ZB}}}\right)}^2\left]{\gamma }_k^2\right\}}^{\frac{1}{2}}$. A comparison between theory and experiment is made for ${\mathrm{K}}_2$Ni${\mathrm{F}}_4$ and the expression for the perpendicular susceptibility of singleion anisotropic antiferromagnets is given.