Approximating macroscopic observables in quantum spin systems with commuting matrices

TL;DR

This paper demonstrates that macroscopic observables in quantum spin systems can be approximated by commuting matrices, despite the general non-approximability of asymptotically commuting matrices in norm.

Contribution

It proves that macroscopic observables in quantum spin systems can be approximated by commuting matrices, a property not true for arbitrary asymptotically commuting matrices.

Findings

Macroscopic observables commute asymptotically as system size grows.

Any set of asymptotically commuting matrices can be approximated by commuting matrices in this context.

The result clarifies the structure of macroscopic observables in quantum spin systems.

Abstract

Macroscopic observables in a quantum spin system are given by sequences of spatial means of local elements $\frac{1}{2n+1}\sum_{j=-n}^n\gamma_j(A_{i}), \; n\in{\mathbb N},\; i=1,...,m$ in a UHF algebra. One of their properties is that they commute asymptotically, as $n$ goes to infinity. It is not true that any given set of asymptotically commuting matrices can be approximated by commuting ones in the norm topology. In this paper, we show that for macroscopic observables, this is true.