  We investigate the role of momentum for the transport of magnetization in the
spin-1/2 Heisenberg chain above the isotropic point at finite temperature and
momentum. Using numerical and analytical approaches, we analyze the
autocorrelations of density and current and observe a finite region of the
Brillouin zone with diffusive dynamics below a cut-off momentum, and a
diffusion constant independent of momentum and time, which scales inversely
with anisotropy. Lowering the temperature over a wide range, starting from
infinity, the diffusion constant is found to increase strongly while the
momentum space cut-off for diffusion decreases. Above the cut-off momentum
diffusion breaks down completely.
