  The Heisenberg spin chain is a canonical integrable model. As such, it
features stable ballistically propagating quasiparticles, but spin transport is
sub-ballistic at any nonzero temperature: an initially localized spin
fluctuation spreads in time $t$ to a width $t^{2/3}$. This exponent, as well as
the functional form of the dynamical spin correlation function, suggest that
spin transport is in the Kardar-Parisi-Zhang (KPZ) universality class. However,
the full counting statistics of magnetization is manifestly incompatible with
KPZ scaling. A simple two-mode hydrodynamic description, derivable from
microscopic principles, captures both the KPZ scaling of the correlation
function and the coarse features of the full counting statistics, but remains
to be numerically validated. These results generalize to any integrable spin
chain invariant under a continuous nonabelian symmetry, and are surprisingly
robust against moderately strong integrability-breaking perturbations that
respect the nonabelian symmetry.
