Spin diffusion of lattice fermions in one dimension

We study long-time spin diffusion of harmonically trapped lattice fermions in one dimension. Combining thermodynamic Bethe ansatz approach and local-density approximation, we calculate spin current and spin-diffusion coefficient driven by the population imbalance. We find spin current is driven by susceptibility effects rather than typical diffusion where magnetization would transport from regions of high magnetization to low. As expected, spin transport is zero through insulating regions and only present in the metallic regions. In the weak-coupling limit, the local spin-diffusion coefficient shows maxima at all the insulating regions. Further, we estimate damping rate of diffusion modes in the weak-coupling limit within the lower metallic portion of the cloud.