Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems

TL;DR

This paper proves that for short-range quantum systems with finite correlation length, the canonical and microcanonical ensembles are approximately equivalent on small regions, using advanced probabilistic and quantum information techniques.

Contribution

It establishes local equivalence of ensembles for non-critical quantum systems with a novel proof approach combining Berry-Esseen bounds and quantum information theory.

Findings

Ensembles are approximately equal on regions of size up to O(N^{1/(d+1)})

The proof applies to systems with finite correlation length at any temperature

Uses a quantum version of the Berry-Esseen theorem

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.