  Although integrable spin chains only host ballistically propagating particles
they can still feature diffusive spin transport. This diffusive spin transport
originates from quasiparticle charge fluctuations inherited from the initial
state's magnetization Gaussian fluctuations. We show that ensembles of initial
states with quasi-long range correlations lead to superdiffusive spin transport
with a tunable dynamical exponent. We substantiate our prediction with
numerical simulations and explain how deviations arise from finite time and
finite size effects.
