Spin-current relaxation time in spin-polarized Heisenberg paramagnets

We study the spatial Fourier transform of the spin correlation function G(q)(t) in paramagnetic quantum, crystals by direct simulation of a 1d lattice of atoms interacting via a nearest-neighbor Heisenberg exchange Hamiltonian. Since it is not practical to diagonalize the s = 1/2 exchange Hamiltonian for a lattice which is of sufficient size to study long-wavelength (hydrodynamic) fluctuations, we instead study the s -> infinity limit and treat each spin as a vector with a classical equation of motion. The simulations give a detailed picture of the correlation function, G(q)(t) and its time derivatives. At high polarization, there seems to be a hierarchy of frequency scales: the local exchange frequency, a wavelength-independent relaxation rate 1/tau that vanishes at large polarization P -> 1, and a wavelength-dependent spin-wave frequency proportional to q(2). This suggests a form for the Correlation function which modifies the spin diffusion coefficients obtained in a moments calculation by Cowan and Mullin, who used a standard Gaussian ansatz for the second derivative of the correlation function.