Kinetic theory of spin superdiffusion in Heisenberg spin chains at high temperature

Sarang Gopalakrishnan, Romain Vasseur

Published 2018-12-06Version 1

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 $D \sim (\Delta - 1)^{-1/2}$.