Symmetry of heat conductivity in inhomogeneous quantum chains

We address the problem of the existence of thermal rectification in inhomogeneous (in particular, graded) chains. Searching for analytical results, we investigate the symmetry properties of the thermal conductivity of the quantum inhomogeneous harmonic chain of oscillators with self-consistent reservoirs, an analytically treatable effective anharmonic model (the inner reservoirs mimic the anharmonic on-site potentials). Considering the linear response regime, i.e., for the system submitted to a small gradient of temperature, we show that, even with quantum effects in the conductivity, there is no thermal rectification. Moreover, as a secondary result, we analyze an old expression derived for the thermal conductivity of pure harmonic chains (i.e., a chain with baths at the boundaries only), and prove that there is no thermal rectification in such inhomogeneous systems, as suggested by numerical simulations in previous works.