Anomalous low-frequency conductivity in easy-plane XXZ spin chains

In the easy-plane regime of XXZ spin chains, spin transport is ballistic, with a Drude weight that has a discontinuous fractal dependence on the value of the anisotropy $Δ= \cos πλ$ at nonzero temperatures. We show that this structure necessarily implies the divergence of the low-frequency conductivity for generic irrational values of $λ$. Within the framework of generalized hydrodynamics, we show that in the high-temperature limit the low-frequency conductivity at a generic anisotropy scales as $σ(ω) \sim 1/\sqrtω$; anomalous response occurs because quasiparticles undergo Lévy flights. For rational values of $λ$, the divergence is cut off at low frequencies and the corrections to ballistic spin transport are diffusive. We also use our approach to recover that at the isotropic point $Δ=1$, spin transport is superdiffusive with $σ(ω) \sim ω^{-1/3}$. We support our results with extensive numerical studies using matrix-product operator methods.

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