Diffusion of solitons in anisotropic Heisenberg models

Abstract

We are interested in the thermal diffusion of a solitary wave in the anisotropic Heisenberg spin chain (HSC) with nearest-neighbor exchange interactions. The shape of the solitary wave is approximated by soliton solutions of the continuum HSC with on-site anisotropy, restricting ourselves to large width excitations. Temperature is simulated by white noise coupled to the system. The noise affects the shape and position of the solitary wave and produces magnons. Using implicit collective variables we describe the former effects and neglect magnons (i.e. we use the so-called adiabatic approximation). We derive stochastic equations of motion for the collective variables which we treat both analytically and numerically. Predictions for the mean values and the variances of the variables obtained from these equations are compared with the corresponding results from spin dynamics simulations. For the soliton position we find reasonable agreement between spin dynamics and the results of the collective variable treatment, whereas we observe deviations for the other collective variables. The stochastic dynamics of the position shows both a standard Brownian and a super-diffusive component. These results are analogous to results for the isotropic case, previously studied by some of the authors. In the present article we discuss in particular how the anisotropy enters the stochastic equations of motion and the quantitative changes it causes to the diffusion.