  Finite-temperature spin transport in the quantum Heisenberg spin chain is
known to be superdiffusive, and has been conjectured to lie in the
Kardar-Parisi-Zhang (KPZ) universality class. Using a kinetic theory of
transport, we compute the KPZ coupling strength for the Heisenberg chain as a
function of temperature, directly from microscopics; the results agree well
with density-matrix renormalization group simulations. We establish a rigorous
quantum-classical correspondence between the "giant quasiparticles" that govern
superdiffusion and solitons in the classical continuous Landau-Lifshitz
ferromagnet. We conclude that KPZ universality has the same origin in classical
and quantum integrable isotropic magnets: a finite-temperature gas of
low-energy classical solitons.
