Transport in the $\mathrm{XXZ}$ model

We present evidence suggesting that spin transport in the gapless phase of the $S=1/2\mathrm{}\hspace{0.5em}\mathrm{XXZ}$ model is ballistic rather than diffusive. We map the model onto a spinless fermion model whose charge stiffness determines the spin transport of the original model. By means of exact numerical diagonalization and finite size scaling we study both the stiffness and the level statistics. We show that the stiffness is nonzero at any temperature so that the transport is ballistic. Our results suggest that the nonzero stiffness arises because even in the presence of umklapp scattering a nonzero fraction of states remains degenerate in the thermodynamic limit.