  We address the hydrodynamics of operator spreading in interacting integrable
lattice models. In these models, operators spread through the ballistic
propagation of quasiparticles, with an operator front whose velocity is locally
set by the fastest quasiparticle velocity. In interacting integrable systems,
this velocity depends on the density of the other quasiparticles, so
equilibrium density fluctuations cause the front to follow a biased random
walk, and therefore to broaden diffusively. Ballistic front propagation and
diffusive front broadening are also generically present in non-integrable
systems in one dimension; thus, although the mechanisms for operator spreading
are distinct in the two cases, these coarse grained measures of the operator
front do not distinguish between the two cases. We present an expression for
the front-broadening rate; we explicitly derive this for a particular
integrable model (the "Floquet-Fredrickson-Andersen" model), and argue on
kinetic grounds that it should apply generally. Our results elucidate the
microscopic mechanism for diffusive corrections to ballistic transport in
interacting integrable models.
