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.