Kardar-Parisi-Zhang universality from soft modes in spin chains

1. Superdiffusion of magnetization in spin chains with isotropic rotational symmetry has now been observed numerically and experimentally in a variety of spin chain models, both at and near integrability. This phenomenon is characterized by a dynamical exponent z=3/2 and has further been conjectured to lie in the Kardar-Parisi-Zhang universality class of dynamics. In this talk we present the current theoretical understanding of how Kardar-Parisi-Zhang universality arises in such models. We argue that this phenomenon originates in the fluctuating hydrodynamics of long-wavelength soft modes, which give rise to a conserved momentum-like degree of freedom satisfying the stochastic Burgers equation.