Soliton diffusion on the classical, isotropic Heisenberg chain

We investigate the diffusive motion of a solitary wave on a classical, isotropic, ferromagnetic Heisenberg spin chain with nearest-neighbour exchange interaction. The spins are coupled magnetically to Gaussian white noise and are subject to Gilbert damping. The noise induces a collective, stochastic time evolution of the solitary wave. Within a continuum version of the model we employ implicit collective variables to describe this stochastic behaviour. Thermally excited magnons are disregarded. We derive stochastic equations of motion for the collective variables and solve them numerically, in particular to obtain their variances as functions of time. These results are compared to data from spin dynamics simulations of a discrete chain. For some of the collective variables we find good agreement with respect to the long time behaviour, whereas for other variables the agreement is only qualitative; reasons for this are given. For shorter times we derive analytical expressions for the variances of the collective variables, which also agree well with spin dynamics.