  For a class of typical states, the real-time and real-space dynamics of
non-equilibrium density profiles has been recently studied for integrable
models, i.e. the spin-1/2 XXZ chain [PRB 95, 035155 (2017)] and the
Fermi-Hubbard chain [PRE 96, 020105 (2017)]. It has been found that the
non-equilibrium dynamics agrees with linear response theory. Moreover, in the
regime of strong interactions, clear signatures of diffusion have been
observed. However, this diffusive behavior strongly depends on the choice of
the initial state and disappears for untypical states without internal
randomness. In the present work, we address the question whether or not the
above findings persist for non-integrable models. As a first step, we study the
spin-1/2 XXZ chain, where integrability can be broken due to an additional
next-nearest neighbor interaction. Furthermore, we analyze the differences of
typical and untypical initial states on the basis of their entanglement and
their local density of states.
