The Lieb-Robinson condition and the Fr\'echet topology

TL;DR

This paper explores different notions of locality in quantum spin systems, showing that the common Lieb-Robinson bound implies but does not equate to continuity in a non-commutative Fréchet topology.

Contribution

It clarifies the relationship between Lieb-Robinson bounds and topological notions of locality in quantum spin systems.

Findings

Lieb-Robinson bounds imply a form of locality but are not equivalent to Fréchet continuity.

The paper introduces a non-commutative Schwartz space analog for quantum observables.

It establishes the distinction between different locality notions in infinite quantum systems.

Abstract

We define various notions of locality for *-automorphisms of the algebra of observables for an infinitely extended quantum spin system and study their relationship. In particular, we show that the ubiquitous characterization which arises from the Lieb-Robinson bound implies but is not equivalent to continuity with respect to the natural Fr\'echet topology of almost local observables, which is a non-commutative analog of the Schwartz space.