Magnetic excitations in vanadium spinels

We study magnetic excitations in vanadium spinel oxides AV$_2$O$_4$ (A=Zn, Mg, Cd) using two models: first one is a superexchange model for vanadium S=1 spins, second one includes in addition spin-orbit coupling, and crystal anisotropy. We show that the experimentally observed magnetic ordering can be obtained in both models, however the orbital ordering is different with and without spin-orbit coupling and crystal anisotropy. We demonstrate that this difference strongly affects the spin-wave excitation spectrum above the magnetically ordered state, and argue that the neutron measurement of such dispersion is a way to distinguish between the two possible orbital orderings in AV$_2$O$_4$.