The Heisenberg spin chain is a standard integrable model. So it features stable ballistic-propagating quasiparticles, but spin transport is quasi-ballistic at non-zero temperatures. That is, the initially localized spin fluctuation spreads out to width $t^{2/3}$ at time $t$. This exponent, and the functional form of the dynamic spin correlation function, suggest that spin transport is in the Kardar-Parisi-Zhang (KPZ) universality class. However, the magnetization full-count statistics are clearly incompatible with KPZ scaling. A simple two-modal hydrodynamic description derived from microscopic principles captures both his KPZ scaling of the correlation function and the coarse features of the full counting statistics, but needs to be verified numerically . These results generalize to any integrable spin-chain invariant under continuous non-Nabelian symmetry and for perturbations that destroy moderately strong integrability respecting non-Nabelian symmetry and surprisingly robust.

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