  Although the Bethe ansatz solution of the spin-1/2 Heisenberg model dates
back nearly a century, the anomalous nature of its high-temperature transport
dynamics has only recently been uncovered. Indeed, numerical and experimental
observations have demonstrated that spin transport in this paradigmatic model
falls into the Kardar-Parisi-Zhang (KPZ) universality class. This has inspired
the significantly stronger conjecture that KPZ dynamics, in fact, occur in all
integrable spin chains with non-Abelian symmetry. Here, we provide extensive
numerical evidence affirming this conjecture. Moreover, we observe that KPZ
transport is even more generic, arising in both supersymmetric and
periodically-driven models. Motivated by recent advances in the realization of
SU(N)-symmetric spin models in alkaline-earth-based optical lattice
experiments, we propose and analyze a protocol to directly investigate the KPZ
scaling function in such systems.
