Stochastic motion of solitary excitations on the classical Heisenberg chain

Abstract

We study stochastic motion of solitary excitations on a classical, discrete, isotropic, ferromagnetic Heisenberg spin chain with nearest-neighbour exchange interactions. Gaussian white noise is coupled to the spins in a way that allows for the noise to be interpreted as a stochastic magnetic field. The noise translates into a collective stochastic force affecting a solitary excitation as a whole. The position of a solitary excitation has to be calculated from the noisy spin configuration, i.e. the position is defined as a function of the spin components. Two examples of such definitions are given, because we want to investigate the dependence of the results on the choice of definition. Using these definitions, we calculate the variance of the position as a function of time and determine the variance from simulations as well. The calculations require knowledge of the shape of the solitary wave. We approximate the shape with that of soliton solutions of the continuum Heisenberg chain, restricting our considerations to solitary waves of large width, in which case this approximation is good. The calculations yield a linear dependence of the variance on time, the slope being determined by parameters describing the shape of the soliton. The two definitions of the position we use provide different results for this slope. The origin of this difference is discussed. With both definitions very good agreement is found between the results of the simulations and the corresponding theoretical results, for not too large time scales.