Local topological order, Haag duality, and reflection positivity
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909033094447104 |
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| author | Naaijkens, Pieter Penneys, David Wallick, Daniel |
| author_facet | Naaijkens, Pieter Penneys, David Wallick, Daniel |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_10693 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local topological order, Haag duality, and reflection positivity Naaijkens, Pieter Penneys, David Wallick, Daniel Mathematical Physics Strongly Correlated Electrons Operator Algebras Quantum Physics 81T05, 81R15 (primary), 18M20, 46L60, 46L99 (secondary) 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. |
| title | Local topological order, Haag duality, and reflection positivity |
| topic | Mathematical Physics Strongly Correlated Electrons Operator Algebras Quantum Physics 81T05, 81R15 (primary), 18M20, 46L60, 46L99 (secondary) |
| url | https://arxiv.org/abs/2605.10693 |