  We study heat transport in a one-dimensional inhomogeneous quantum spin 1/2
system. It consists of a finite-size XX spin chain coupled at its ends to
semi-infinite XX and XY chains at different temperatures, which play the role
of heat and spin reservoirs. After using the Jordan-Wigner transformation we
map the original spin Hamiltonian into a fermionic Hamiltonian, which contains
normal and pairing terms. We find the expressions for the heat currents and
solve the problem with a non-equilibrium Green's function formalism. We analyze
the behavior of the heat currents as functions of the model parameters. When
finite magnetic fields are applied at the two reservoirs, the system exhibits
rectification effects in the heat flow.
