  We study the nonequilibrium dynamics of random spin chains that remain
integrable (i.e., solvable via Bethe ansatz): because of correlations in the
disorder, these systems escape localization and feature ballistically spreading
quasiparticles. We derive a generalized hydrodynamic theory for dynamics in
such random integrable systems, including diffusive corrections due to
disorder, and use it to study non-equilibrium energy and spin transport. We
show that diffusive corrections to the ballistic propagation of quasiparticles
can arise even in noninteracting settings, in sharp contrast with clean
integrable systems. This implies that operator fronts broaden diffusively in
random integrable systems. By tuning parameters in the disorder distribution,
one can drive this model through an unusual phase transition, between a phase
where all wavefunctions are delocalized and a phase in which low-energy
wavefunctions are quasi-localized (in a sense we specify). Both phases have
ballistic transport; however, in the quasi-localized phase, local
autocorrelation functions decay with an anomalous power law, and the density of
states diverges at low energy.
