TY - JOUR

T1 - Gaussian concentration bound and Ensemble equivalence in generic quantum many-body systems including long-range interactions

AU - Kuwahara, Tomotaka

AU - Saito, Keiji

N1 - Funding Information:
The work of T. K. was supported by the RIKEN Center for AIP and JSPS KAKENHI Grant No. 18K13475 . K.S. was supported by JSPS Grants-in-Aid for Scientific Research (Grants No. JP16H02211 , No. JP19H05791 , and No. JP19H05603 ).
Publisher Copyright:
© 2020

PY - 2020/10

Y1 - 2020/10

N2 - 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.

AB - 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.

KW - Chernoff–Hoeffding inequality

KW - Concentration bound

KW - Eigenstate thermalization hypothesis

KW - Ensemble equivalence

KW - Long-range-interacting systems

KW - Weak eigenstate thermalization

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U2 - 10.1016/j.aop.2020.168278

DO - 10.1016/j.aop.2020.168278

M3 - Article

AN - SCOPUS:85089498628

VL - 421

JO - Annals of Physics

JF - Annals of Physics

SN - 0003-4916

M1 - 168278

ER -