  In this work, with focus on the energy transport properties in quantum, low
dimensional, graded materials, we address the investigation of the energy (and
spin) current in XXZ open chains with graded inner structures and driven out of
equilibrium by magnetization pumping applied at the ends. We study several
types of graded structures in different situations in order to show a
ubiquitous occurrence of energy rectification, even for the system under a
homogeneous magnetic field. Due to technical difficulties, we carry out the
computation for small chains, but we present arguments which indicate the
extension of some results to larger systems. Recalling the generic existence of
energy rectification in classical, graded materials, which are described by
anharmonic chains of oscillators, and recalling also the anharmonicity of these
XXZ models, which involve quartic terms in more transparent representation in
terms of fermionic creation and annihilation operators, we may say that our
results extend the ubiquity of energy rectification occurrence in classical
graded materials to the case of quantum systems.
