  Equilibrium spatio-temporal correlation functions are central to
understanding weak nonequilibrium physics. In certain local one-dimensional
classical systems with three conservation laws they show universal features.
Namely, fluctuations around ballistically propagating sound modes can be
described by the celebrated Kardar-Parisi-Zhang (KPZ) universality class. Can
such universality class be found also in quantum systems? By unambiguously
demonstrating that the KPZ scaling function describes magnetization dynamics in
the SU(2) symmetric Heisenberg spin chain we show, for the first time, that
this is so. We achieve that by introducing new theoretical and numerical tools,
and make a puzzling observation that the conservation of energy does not seem
to matter for the KPZ physics.
