Local topological order, Haag duality, and reflection positivity

• URL: http://arxiv.org/abs/2605.10693v1
• Date: Mon, 11 May 2026 15:08:43 GMT
• Title: Local topological order, Haag duality, and reflection positivity
• Abstract summary: We introduce an axiom for LTOs which ensures Haag duality for cone-like regions using Tomita-Takesaki theory.<n>We prove this axiom is satisfied for all known topologically ordered commuting projector models.<n>We also give a reflection positivity axiom for LTOs, connecting to the recent article.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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