Thermal transport through junctions of critical spin chains

We investigate the steady-state energy transport through a system of two coupled 1D spin chains at different temperatures. In the case of two identical spin chains we compare our results to the known expression J_E(T_L,T_R) = cπ/12 (T_L²-T_R²) for gapless chains with a conformal charge c. In addition we also consider a system of two different spin chains, where the situation is less clear.