Magnetic excitations in rare-earth antiferromagnets with bilinear and biquadratic pair couplings: Application to DySb

A Green's-function formalism is constructed for the purpose of computing the elementary excitation energies in a type-II antiferromagnet with Heisenberg exchange and quadrupolar couplings in a cubic crystal field. Three types of excitation modes are found: a longitudinal mode ($L$ mode) associated with $O_0^1$ and $O_0^2$ operators ($\Delta m=0\mathrm{}$), a transverse mode ($T1\mathrm{}$ mode) associated with $O_{\pm 1}^1$ and $O_{\pm 1}^2$ operators ($\Delta m=\pm 1$), and a second transverse mode ($T2\mathrm{}$ mode) associated with $O_{\pm 2}^2$ operators ($\Delta m=\pm 2$). In the ordered phase the $L$-mode and $T1\mathrm{}$-mode excitations are mixed magnetic dipolar and quadrupolar excitations. In the disordered phase, as a consequence of cubic symmetry, the magnetic dipolar modes decouple from the quadrupolar modes, giving rise to the possibility of observing a pure quadrupolar excitation. Cubic symmetry also demands that in the disordered phase certain of the excitation energies in the $L$, $T1\mathrm{}$, and $T2\mathrm{}$ modes have identical dispersion curves. In general, the dispersion in both the ordered and disordered phase is complicated owing to the inclusion of next-nearest-neighbor coupling. The theory is applied to DySb, a type-II antiferromagnet with strong evidences of quadrupolar coupling.