On which length scales can temperature exist in quantum systems?

We consider a regular chain of elementary quantum systems with nearest neighbor interactions and assume that the total system is in a canonical state with temperature T. We analyze under what condition the state factors into a product of canonical density matrices with respect to groups of n subsystems each, and when these groups have the same temperature T. While in classical mechanics the validity of this procedure only depends on the size of the groups n, in quantum mechanics the minimum group size n_min also depends on the temperature T! As examples, we apply our analysis to different types of Heisenberg spin chains.