  Transport phenomena are central to physics, and transport in the many-body
and fully-quantum regime is attracting an increasing amount of attention. It
has been recently revealed that some quantum spin chains support ballistic
transport of excitations at all energies. However, when joining two
semi-infinite ballistic parts, such as the XX and XXZ spin-1/2 models, our
understanding suddenly becomes less established. Employing a
matrix-product-state ansatz of the wavefunction, we study the relaxation
dynamics in this latter case. Here we show that it takes place inside a light
cone, within which two qualitatively different regions coexist: an inner one
with a strong tendency towards thermalization, and an outer one supporting
ballistic transport. We comment on the possibility that even at infinite time
the system supports stationary currents and displays a non-zero Kapitza
boundary resistance. Our study paves the way to the analysis of the interplay
between transport, integrability, and local defects.
