Gaussian concentration bound and Ensemble equivalence in generic quantum many-body systems including long-range interactions

Tomotaka Kuwahara, Keiji Saito

Research output: Contribution to journalArticle

Abstract

This work explores fundamental statistical and thermodynamic properties of short-and long-range-interacting systems. The purpose of this study is twofold. Firstly, we rigorously prove that the probability distribution of arbitrary few-body observables is restricted by a Gaussian concentration bound (or Chernoff–Hoeffding inequality) above some threshold temperature. This bound is then derived for arbitrary Gibbs states of systems that include long-range interactions Secondly, we establish a quantitative relationship between the concentration bound of the Gibbs state and the equivalence of canonical and micro-canonical ensembles. We then evaluate the difference in the averages of thermodynamic properties between the canonical and the micro-canonical ensembles. Under the assumption of the Gaussian concentration bound on the canonical ensemble, the difference between the ensemble descriptions is upper-bounded by n−1log(n3∕2Δ−1)1∕2 with n being the system size and Δ being the width of the energy shell of the micro-canonical ensemble This limit gives a non-trivial upper bound exponentially small energy width with respect to the system size. By combining these two results, we prove the ensemble equivalence as well as the weak eigenstate thermalization in arbitrary long-range-interacting systems above a threshold temperature.

Original languageEnglish
Article number168278
JournalAnnals of Physics
Volume421
DOIs
Publication statusPublished - 2020 Oct

Keywords

  • Chernoff–Hoeffding inequality
  • Concentration bound
  • Eigenstate thermalization hypothesis
  • Ensemble equivalence
  • Long-range-interacting systems
  • Weak eigenstate thermalization

ASJC Scopus subject areas

  • Physics and Astronomy(all)

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