Kinetic theory of spin superdiffusion in Heisenberg spin chains at high temperature. (arXiv:1812.02701v1 [cond-mat.stat-mech])

We address the nature of spin transport in the integrable XXZ spin chain,
focusing on the isotropic Heisenberg limit. We calculate the diffusion constant
using a kinetic picture based on generalized hydrodynamics: we find that it
diverges, and show that a self-consistent treatment of this divergence gives
superdiffusion, with an effective time-dependent diffusion constant that scales
as $D(t) \sim t^{1/3} \log^{2/3} t$. This exponent had previously been observed
in large-scale numerical simulations, but had not been theoretically explained.
Our results also make clear why the anomalous diffusion in the present case is
a qualitatively different phenomenon from Levy flights and other phenomena in
random systems. We briefly discuss XXZ models with easy-axis anisotropy $\Delta
> 1$; for these our treatment predicts diffusion, with a diffusion constant $D$
that saturates at large anisotropy, and diverges as the Heisenberg limit is
approached, as $D \sim (\Delta - 1)^{-1/2}$.

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