  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 combined with
Gaussian fluctuations: 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 $D(t) \sim t^{1/3}$.
This exponent had previously been observed in large-scale numerical
simulations, but had not been theoretically explained. We briefly discuss XXZ
models with easy-axis anisotropy $\Delta > 1$. Our method gives closed-form
expressions for the diffusion constant $D$ in the infinite-temperature limit
for all $\Delta > 1$. We find that $D$ saturates at large anisotropy, and
diverges as the Heisenberg limit is approached, as $D \sim (\Delta -
1)^{-1/2}$.
