Propagation of spin waves

A mutilated model is constructed to approximate the collision term of spin Boltzmann equation that incorporates newly appearing collisional invariants i.e, the total angular momentum. With recourse to degenerate perturbation theory, the dispersion relations of hydrodynamic modes are formulated, among which spin modes are responsible for spin equilibration. We find that the non-locality does not change the sound speed but slows down the propagation of spin waves. The damping rates of spin modes are close to those of spinless modes over a reasonable parameter value range. The results reveal that both spin and momentum should be treated simultaneously in a unified transport framework. In the nonrelativistic limit, the short-wavelength behavior for normal modes is also explored and there exists a critical point for every distinct discrete mode over which only quasiparticle modes contribute