Spin Diffusion in the Heisenberg Magnets at Infinite Temperature

A theoretical criterion for the occurrence of the spin diffusion is given in terms of the "friction function" or "memory function" which occurs in the kinetic equation of the Fourierspace transform $I(\overrightarrow{\mathrm{k}}, t)$ of the two-time spin-pair correlation function. The friction function in the limit of $\overrightarrow{\mathrm{k}}\to 0$ is determined with the aid of the short-time expansion of the two-time spin-pair correlation function for the square, sc, and bcc isotropic Heisenberg magnets at infinite temperature, and the criterion is checked. The spin diffusion constants for spin ½ are $0.860J{a}^2$, $0.619J{a}^2$, and $0.509J{a}^2$, respectively, for these lattices, where $J$ is the exchange integral and $a$ is the lattice constant. The spin diffusion constants for larger spins are also provided. It should be noted that $0.509J{a}^2$ is in complete agreement with the experimental value $(0.525\mathrm{}\pm 0.06)J{a}^2$ for the bcc solid ${\mathrm{He}}^3$. For the one-dimensional isotropic Heisenberg magnet and for the square, sc, and bcc $\mathrm{XY}$ magnets, the friction function could not be determined in the present calculation. Suggested values of the spin diffusion constant are given for these lattices. Comparison is made with the results of Windsor's computer-simulation calculation. There exists no spin diffusion for the one-dimensional $\mathrm{XY}$ model.