Lieb-Robinson Bound at Finite Temperature

The Lieb-Robinson bound shows that the speed of propagating information in a nonrelativistic quantum lattice system is bounded by a finite velocity, which entails the clustering of correlations. In this paper, we extend the Lieb-Robinson bound to quantum systems at finite temperature by calculating the dynamical correlation function at nonzero temperature for systems whose interactions are respectively short-range, exponentially-decaying and long-range. We introduce a simple way of counting the clusters in a cluster expansion by using the combinatoric generating functions of graphs. Limitations and possible applications of the obtained bound are also discussed.