Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems

Abstract

We consider the problem of whether the canonical and microcanonical ensembles are locally equivalent for short-ranged quantum Hamiltonians of $N$ spins arranged on a $d$-dimensional lattices. For any temperature for which the system has a finite correlation length, we prove that the canonical and microcanonical state are approximately equal on regions containing up to $O(N^{1/(d+1)})$ spins. The proof rests on a variant of the Berry--Esseen theorem for quantum lattice systems and ideas from quantum information theory.

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• Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature
• Most energetic passive states
• Finite-size scaling of out-of-time-ordered correlators at late times
• Thermalization and Return to Equilibrium on Finite Quantum Lattice Systems
• Optimal Quantum Thermometry with Coarse-grained Measurements
• Work and entropy production in generalised Gibbs ensembles
• Concentration bounds for quantum states with finite correlation length on quantum spin lattice systems
• Eigenstate thermalization from the clustering property of correlation
• Operational axioms for diagonalizing states
• Quantum concentration inequalities
• The modified logarithmic Sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions
• Convergence rates for the quantum central limit theorem
• Extremal eigenvalues of local Hamiltonians
• Minimal Effective Gibbs Ansatz (MEGA): A simple protocol for extracting an accurate thermal representation for quantum simulation
• Thermalization of locally perturbed many-body quantum systems
• Finite-size scaling analysis of eigenstate thermalization
• Instability of localization in translation-invariant systems
• Matrix product state approximations to quantum states of low energy variance
• Convergence of eigenstate expectation values with system size
• Deviation from maximal entanglement for mid-spectrum eigenstates of local Hamiltonians
• Theoretical study on thermalization in isolated quantum systems