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

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 ${\left[n^{-1}\log (n^{3/2}{\Delta }^{-1})\right]}^{1/2}$ with $n$ being the system size and $\Delta$ being the width of the energy shell of the micro-canonical ensemble This limit gives a non-trivial upper bound \textit{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.

28 pages, 6 figures

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Cited by in corpus (28)

• Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature
• Improved thermal area law and quasi-linear time algorithm for quantum Gibbs states
• Classical simulation of short-time quantum dynamics
• Quantum many-body systems in thermal equilibrium
• Eigenstate thermalization from the clustering property of correlation
• Quantum concentration inequalities
• Learning quantum many-body systems from a few copies
• Rapid thermalization of dissipative many-body dynamics of commuting Hamiltonians
• Thermalization of locally perturbed many-body quantum systems
• Localized Virtual Purification
• Energy measurements remain thermometrically optimal beyond weak coupling
• Concentration bounds for quantum states and limitations on the QAOA from polynomial approximations
• Thermal Area Law in Long-Range Interacting Systems
• Clustering theorem in 1D long-range interacting systems at arbitrary temperatures
• Thermal Area Law for Lattice Bosons
• Limitations of variational quantum algorithms: a quantum optimal transport approach
• Long-time equilibration can determine transient thermality
• On the fluctuations of the number of atoms in the condensate
• Symmetry-prohibited thermalization after a quantum quench
• Quantum thermalization must occur in translation-invariant systems at high temperature
• Allosteric impurity effects in long spin chains
• Quantum concentration inequalities and equivalence of the thermodynamical ensembles: an optimal mass transport approach
• Energy diffusion in the long-range interacting spin systems
• Thermalization in a simple spin-chain model
• High-temperature partition functions and classical simulatability of long-range quantum systems
• Basis dependence of eigenstate thermalization
• Absence of thermalization after a local quench and strong violation of the eigenstate thermalization hypothesis
• Temperature and conditions for thermalization after canonical quenches