On the local equivalence between the canonical and the microcanonical distributions for quantum spin systems

TL;DR

This paper proves that for large quantum spin systems, the local expectation values in the canonical and microcanonical ensembles are close under certain conditions, improving previous estimates with elementary methods.

Contribution

It establishes local ensemble equivalence for quantum spin systems with improved estimates and elementary proofs, extending prior results.

Findings

Local expectation values in both ensembles are close for large systems.

The results hold for small inverse temperature and small support regions.

Provides elementary proofs and reviews standard thermodynamic limits.

Abstract

We study a quantum spin system on the $d$-dimensional hypercubic lattice $\Lambda$ with $N=L^d$ sites with periodic boundary conditions. We take an arbitrary translation invariant short-ranged Hamiltonian. For this system, we consider both the canonical ensemble with inverse temperature $\beta_0$ and the microcanonical ensemble with the corresponding energy $U_N(\beta_0)$. For an arbitrary self-adjoint operator $\hat{A}$ whose support is contained in a hypercubic block $B$ inside $\Lambda$, we prove that the expectation values of $\hat{A}$ with respect to these two ensembles are close to each other for large $N$ provided that $\beta_0$ is sufficiently small and the number of sites in $B$ is $o(N^{1/2})$. This establishes the equivalence of ensembles on the level of local states in a large but finite system. The result is essentially that of Brandao and Cramer (here restricted to the case of the canonical and the microcanonical ensembles), but we prove improved estimates in an elementary manner. We also review and prove standard results on the thermodynamic limits of thermodynamic functions and the equivalence of ensembles in terms of thermodynamic functions. The present paper assumes only elementary knowledge on quantum statistical mechanics and quantum spin systems.