Local topological order, Haag duality, and reflection positivity

TL;DR

This paper introduces axioms for local topological order that ensure Haag duality and reflection positivity in quantum spin systems, providing new proofs and connections to known models and recent theories.

Contribution

It establishes axioms guaranteeing Haag duality and reflection positivity for topologically ordered models, extending the theoretical framework and confirming these properties for known models.

Findings

Proves Haag duality for all known topologically ordered commuting projector models.

Introduces reflection positivity axiom satisfied by models with Z/2 symmetry.

Provides an independent proof of Haag duality for Levin-Wen string net models.

Abstract

In our previous article [arXiv:2307.12552], we introduced local topological order (LTO) axioms for abstract quantum spin systems which allow one to access topological order via a boundary algebra construction. Using the LTO axioms, we produced a canonical pure state on the quasi-local algebra, which gives a net of von Neumann algebras associated to a poset of cones in $\mathbb{R}^n$. In this article, motivated by [arXiv:2509.23734], we introduce an axiom for LTOs which ensures Haag duality for cone-like regions using Tomita-Takesaki theory. We prove this axiom is satisfied for all known topologically ordered commuting projector models. We thus get an independent proof of Haag duality for the Levin-Wen string net models originally proved in [arXiv:2509.23734]. We also give a reflection positivity axiom for LTOs, connecting to the recent article [arXiv:2510.20662]. We again prove this axiom is satisfied for all known topologically ordered commuting projector models about some $\mathbb{Z}/2$-reflection symmetry.