Using the framework of generalized hydrodynamics, we compute the low-frequency spin conductivity $\sigma \left(\omega \right)$ of XXZ spin chains with easy-plane anisotropy. We find that for almost all values of the anisotropy $-1<\mathrm{\Delta }<1$, the low-frequency conductivity scales anomalously with frequency, as $\sigma \left(\omega \right)\sim 1/\sqrt{\omega }$. We interpret this anomalous response as a consequence of quasiparticles undergoing Lévy flights. For special values of the anisotropy, the divergence is cut off at low frequencies, so $\sigma \left(\omega \right)$ has a finite dc limit. These results reveal a hitherto unknown mechanism for anomalous response in integrable systems and also provide a physical explanation of the discontinuous behavior of the spin Drude weight. We use our approach to recover that at the isotropic point $\mathrm{\Delta }=1$, $\sigma \left(\omega \right)\sim {\omega }^{-1/3}$. We support our results with extensive numerical studies using matrix-product operator methods.

• Received 20 September 2019
• Revised 23 January 2020
• Accepted 12 May 2020

DOI:https://doi.org/10.1103/PhysRevB.101.224415