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.

The observed acceleration of the present universe is shown to be well explained by the holographic dark energy characterized by the total comoving horizon of the universe (ηHDE). It is of interest to notice that the very large primordial part of the comoving horizon generated by the inflation of the early universe makes the ηHDE behave like a cosmological constant. As a consequence, both the fine-tuning problem and the coincidence problem can reasonably be understood with the inflationary universe and holographical principle. We present a systematic analysis and obtain a consistent cosmological constraint on the ηHDE model based on the recent cosmological observations. It is found that the ηHDE model gives the best-fit result Ω m0 = 0.270 (Ω de0 = 0.730) and the minimal χ 2 min = 542.915 which is compatible with χ 2 ΛCDM = 542.919 for the ΛCDM model.