  In Phys. Rev. Lett. 120, 200603 (2018), a segmented XXZ spin chain with zero
anisotropy in one half and a large anisotropy on the other half gave rise to a
spin current rectification which is perfect in the thermodynamic limit. Here we
extend the previous study to segmented chains with interacting integrable as
well as non-integrable halves, considering even cases in which no ballistic
transport can emerge in either half. We demonstrate that, also in this more
general case, it is possible to obtain giant rectification when the two
interacting half chains are sufficiently different. We also show that the
mechanism causing this effect is the emergence of an energy gap in the
excitation spectrum of the out-of-equilibrium insulating steady state in one of
the two biases. Finally we demonstrate that in the thermodynamic limit there is
no perfect rectification when each of the two half chains is interacting.
