We study magnetization transport in anisotropic spin-$\frac{1}{2}$ chains governed by the integrable XXZ model with and without integrability-breaking perturbations at high temperatures $\left(T\to \infty \right)$ using a hybrid approach that combines exact sum rules with judiciously chosen Ansätze. In the integrable XXZ model we find (i) superdiffusion at the isotropic (Heisenberg) point, with frequency dependent conductivity ${\sigma }^{\prime }(\omega \to 0)\sim {\left|\omega \right|}^{\alpha }$, where $\alpha =-3/7$ in close numerical agreement with recent $t$-DMRG computations; (ii) a continuously drifting exponent from $\alpha =-1^{+}$ in the XY (gapless) limit of the model to $\alpha >0$ within the Ising (gapped) regime; and (iii) a diffusion constant saturating in the XY coupling deep in the Ising limit. We consider two kinds of next-nearest-neighbor integrability breaking perturbations—a simple spin-flip term $\left(J_2\right)$ and a three-spin assisted variant $\left(t_2\right)$, natural in the fermion particle representation of the spin chain. In the first case we discover a remarkable sensitivity of ${\sigma }^{\prime }\left(\omega \right)$ to the sign of $J_2$, with enhanced low frequency spectral weight and a pronounced upward shift in the magnitude of $\alpha$ for $J_2>0$. Perhaps even more surprising, we find subdiffusion $\left(\alpha >0\right)$ over a range of $J_2<0$. By contrast, the effects of the “fermionic” three-spin perturbation are sign symmetric; this perturbation produces a clearly observable hydrodynamic relaxation. At large strength of the integrability breaking term $J_2\to \pm \infty$ the problem is effectively noninteracting (fermions hopping on odd and even sublattices) and we find $\alpha \to -1$ behavior reminiscent of the XY limit of the integrable XXZ chain. Exact diagonalization studies largely corroborate these findings over accessible frequencies.

• Received 7 March 2018
• Revised 27 June 2018

DOI:https://doi.org/10.1103/PhysRevB.98.054415