Thouless number and spin diffusion in quantum Heisenberg ferromagnets

Using an analogy between the conductivity tensor of electronic systems and the spin stiffness tensor of spin systems, we introduce the concept of the Thouless number $g_0$ and the dimensionless frequency dependent conductance $g( \omega )$ for quantum spin models. It is shown that spin diffusion implies the vanishing of the Drude peak of $g ( \omega )$, and that the spin diffusion coefficient $D_s$ is proportional to $g_0$. We develop a new method based on the Thouless number to calculate $D_s$, and present results for $D_s$ in the nearest-neighbor quantum Heisenberg ferromagnet at infinite temperatures for arbitrary dimension $d$ and spin $S$.