  The emergence of superdiffusive spin dynamics in integrable classical and
quantum magnets is well established by now, but there is no generally valid
theoretical explanation for this phenomenon. A fundamental difficulty is that
the hydrodynamic fluctuations of conserved quasiparticle modes are purely
diffusive. We argue that in isotropic integrable magnets, a complete
hydrodynamic description must include soft "gauge" degrees of freedom, that
arise from spontaneous breaking of the Bethe pseudovacuum symmetry. We show
that the coarse-grained time evolution of these modes lies in the
Kardar-Parisi-Zhang universality class of dynamics.
