Dynamics and relaxation in spin nematics

We study dynamics and relaxation of elementary excitations (magnons) in the spin nematic (quadrupole ordered) phase of S=1 magnets. We develop a general phenomenological theory of spin dynamics and relaxation for spin-1 systems. Results of the phenomenological approach are compared to those obtained by microscopic calculations for the specific S=1 model with isotropic bilinear and biquadratic exchange interactions. This model exhibits a rich behavior depending on the ratio of bilinear and biquadratic exchange constants, including several points with an enhanced symmetry. It is shown that symmetry plays an important role in relaxation. Particularly, at the SU(3) ferromagnetic point the magnon damping $Γ$ depends on its wavevector k as $Γ\propto k^{4}$, while a deviation from the high-symmetry point changes the behavior of the leading term to $Γ\propto k^{2}$. We point out a similarity between the behavior of magnon relaxation in spin nematics to that in an isotropic ferromagnet.