Stochastic description of the dynamics of the random-exchange Heisenberg chain

We study the diffusion process in a Heisenberg chain with correlated spatial disorder, with a power spectrum in the momentum space behaving as $k^{-\beta }$, using a stochastic description. It establishes a direct connection between the fluctuation in the spin-wave density of states and the noise density of states. For continuous ranges of the exponent $\beta$, we find superdiffusive and ballistic spin-wave motions. Both diffusion exponents predicted by the stochastic procedure agree with the ones calculated using the Hamiltonian dynamics.