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The Lieb-Robinson condition and the Fréchet topology

Sven Bachmann, Giuseppe De Nittis, Julián Gómez

TL;DR

The paper clarifies how various locality notions for automorphisms of the quasi-local algebra of observables relate to the natural Fréchet topology on almost-local observables. It shows that Lieb-Robinson type locality implies Fréchet continuity, but the reverse does not hold, and introduces two equivalent notions of locality-preserving automorphisms (ALP) that coincide with LR-type dynamics. It further establishes that ALP automorphisms restrict to continuous automorphisms of the Fréchet algebra $\mathscr{A}_\infty$ and form a group, while exploring the polynomial-decay regime where LR-type bounds and ALP concepts interact in a dimension-dependent way. The analysis includes explicit constructions with flip and shift automorphisms, a one-parameter group bridging non-ALP cases to the identity, and a detailed treatment of polynomial decays, highlighting when continuity, locality preservation, and group structure align or diverge. Overall, the work provides a rigorous, multi-topology perspective on locality in quantum spin lattices and the associated automorphism dynamics.

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 Fréchet topology of almost local observables, which is a non-commutative analog of the Schwartz space.

The Lieb-Robinson condition and the Fréchet topology

TL;DR

The paper clarifies how various locality notions for automorphisms of the quasi-local algebra of observables relate to the natural Fréchet topology on almost-local observables. It shows that Lieb-Robinson type locality implies Fréchet continuity, but the reverse does not hold, and introduces two equivalent notions of locality-preserving automorphisms (ALP) that coincide with LR-type dynamics. It further establishes that ALP automorphisms restrict to continuous automorphisms of the Fréchet algebra and form a group, while exploring the polynomial-decay regime where LR-type bounds and ALP concepts interact in a dimension-dependent way. The analysis includes explicit constructions with flip and shift automorphisms, a one-parameter group bridging non-ALP cases to the identity, and a detailed treatment of polynomial decays, highlighting when continuity, locality preservation, and group structure align or diverge. Overall, the work provides a rigorous, multi-topology perspective on locality in quantum spin lattices and the associated automorphism dynamics.

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 Fréchet topology of almost local observables, which is a non-commutative analog of the Schwartz space.

Paper Structure

This paper contains 15 sections, 27 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Algebraic description of a spin lattice system
  3. Local and quasi-local observables
  4. Almost-local observables
  5. Automorphisms and Fréchet-type automorphisms
  6. Restriction of automorphisms to the Fréchet $\ast$-algebra
  7. Almost locality preserving automorphisms
  8. Fréchet continuous but not ALP automorphisms
  9. Approximately locality preserving automorphisms
  10. Lieb-Robinson type automorphisms
  11. The Polynomial case
  12. Basic facts on Frechét $\ast$-algebras
  13. Frechét spaces
  14. Frechét $\ast$-algebras
  15. Rapidly decaying sequence

Key Result

Theorem 1.1

Any Lieb-Robinson-type automorphism is continuous on the algebra of almost-local observables with respect to the Fréchet topology. However, Fréchet continuity alone does not imply the existence of a Lieb-Robinson bound.

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • Remark 2.3: Pauli basis
  • Example 2.4: Flip-type automorphisms
  • Example 2.5: Shift-type automorphisms
  • Remark 2.6
  • Proposition 2.7
  • Definition 3.1: Almost locality preserving automorphisms
  • ...and 50 more