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arXiv:2307.12552 · Forum of Mathematics Sigma

Local topological order and boundary algebras

Corey Jones, Pieter Naaijkens, David Penneys, Daniel Wallick, Masaki Izumi

Published Jan 1, 2025• 5 citations
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Abstract

Abstract We introduce a set of axioms for locally topologically ordered quantum spin systems in terms of nets of local ground state projections, and we show they are satisfied by Kitaev’s Toric Code and Levin-Wen type models. For a locally topologically ordered spin system on $\mathbb {Z}^{k}$ , we define a local net of boundary algebras on $\mathbb {Z}^{k-1}$ , which provides a mathematically precise algebraic description of the holographic dual of the bulk topological order. We construct a canonical quantum channel so that states on the boundary quasi-local algebra parameterize bulk-boundary states without reference to a boundary Hamiltonian. As a corollary, we obtain a new proof of a recent result of Ogata [Oga24] that the bulk cone von Neumann algebra in the Toric Code is of type $\mathrm {II}$ , and we show that Levin-Wen models can have cone algebras of type $\mathrm {III}$ . Finally, we argue that the braided tensor category of DHR bimodules for the net of boundary algebras characterizes the bulk topological order in (2+1)D, and can also be used to characterize the topological order of boundary states.

DOI
10.1017/fms.2025.16

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Abstract We introduce a set of axioms for locally topologically ordered quantum spin systems in terms of nets of local ground state projections, and we show they are satisfied by Kitaev’s Toric Code and Levin-Wen type models. For a locally topologically ordered spin system on Zk\mathbb {Z}^{k}Zk , we define a local net of boundary algebras on Zk−1\mathbb {Z}^{k-1}Zk−1 , which provides a mathematically precise algebraic description of the holographic dual of the bulk topological order. We construct a canonical quantum channel so that states on the boundary quasi-local algebra parameterize bulk-boundary states without reference to a boundary Hamiltonian. As a corollary, we obtain a new proof of a recent result of Ogata [Oga24] that the bulk cone von Neumann algebra in the Toric Code is of type II\mathrm {II}II , and we show that Levin-Wen models can have cone algebras of type III\mathrm {III}III . Finally, we argue that the braided tensor category of DHR bimodules for the net of boundary algebras characterizes the bulk topological order in (2+1)D, and can also be used to characterize the topological order of boundary states.
In our previous article [arXiv:2307.12552], we introduced local topological order (LTO) axioms for quantum spin systems which allowed us to define a physical boundary (associated to a cut of the lattice) manifested by a net of boundary algebras in one dimension lower. This gives a formal setting for topological holography, where the braided tensor category of DHR bimodules of the physical boundary algebra captures the bulk topological order. In this article, we extend the LTO axioms to quantum spin systems equipped with a topological boundary (domain wall with the trivial phase), again producing a physical boundary algebra for the bulk-boundary system, whose category of (topological) boundary DHR bimodules recovers the topological boundary order. We perform this analysis in explicit detail for Levin-Wen and Walker-Wang bulk-boundary systems. Along the way, we introduce a 2D braided categorical net of algebras built from a unitary braided fusion category (UBFC). Such nets arise as boundary algebras of Walker-Wang models. We consider the canonical state on this braided categorical net corresponding to the standard topological boundary for the Walker-Wang model. Interestingly, in this state, the cone von Neumann algebras are type I with finite dimensional centers, in contrast with the type II and III cone von Neumann algebras from the Levin-Wen models studied in [arXiv:2307.12552]. The superselection sectors recover the underlying unitary category of our UBFC, and it was recently proven in [arXiv:2609.20725] that the superselection category also captures the fusion and braiding.
The recent article [arXiv:2307.12552] gave local topological order (LTO) axioms for a quantum spin system, showed they held in Kitaev's Toric Code and in Levin-Wen string net models, and gave a bulk boundary correspondence to describe bulk excitations in terms of the boundary net of algebras. In this article, we prove the LTO axioms for Kitaev's Quantum Double model for a finite group GGG. We identify the boundary nets of algebras with fusion categorical nets associated to (Hilb(G),C[G])(\mathsf{Hilb}(G),\mathbb{C}[G])(Hilb(G),C[G]) or (Rep(G),CG)(\mathsf{Rep}(G),\mathbb{C}^G)(Rep(G),CG) depending on whether the boundary cut is rough or smooth respectively. This allows us to make connections to work of Ogata on the type of the cone von Neumann algebras in the algebraic quantum field theory approach to topological superselection sectors. We show that the boundary algebras can also be calculated from a trivial GGG-symmetry protected topological phase (GGG-SPT), and that the gauging map preserves the boundary algebras. Finally, we compute the boundary algebras for the (3+1)D Quantum Double model associated to an abelian group.
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 Rn\mathbb{R}^nRn. 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 Z/2\mathbb{Z}/2Z/2-reflection symmetry.
The recent article by Jones et al. [arXiv:2307.12552 (2023)] gave local topological order (LTO) axioms for a quantum spin system, showed they held in Kitaev’s Toric Code and in Levin-Wen string net models, and gave a bulk boundary correspondence to describe bulk excitations in terms of the boundary net of algebras. In this article, we prove the LTO axioms for Kitaev’s Quantum Double model for a finite group G. We identify the boundary nets of algebras with fusion categorical nets associated to (Hilb(G),C[G]) or (Rep(G),CG) depending on whether the boundary cut is rough or smooth, respectively. This allows us to make connections to the work of Ogata [Ann. Henri Poincaré 25, 2353–2387 (2024)] on the type of the cone von Neumann algebras in the algebraic quantum field theory approach to topological superselection sectors. We show that the boundary algebras can also be calculated from a trivial G-symmetry protected topological phase (G-SPT), and that the gauging map preserves the boundary algebras. Finally, we compute the boundary algebras for the (3 + 1)D Quantum Double model associated to an Abelian group.
We introduce the concept of boundary degeneracy, as the ground state degeneracy of topologically ordered states on a compact orientable spatial manifold with gapped boundaries. We emphasize that the boundary degeneracy provides richer information than the bulk degeneracy. Beyond the bulk-edge correspondence, we find the ground state degeneracy of the fully gapped edge modes depends on boundary gapping conditions. By associating different types of boundary gapping conditions as different ways of particle or quasiparticle condensations on the boundary, we develop an analytic theory of gapped boundaries. By Chern-Simons theory, this allows us to derive the ground state degeneracy formula in terms of boundary gapping conditions, which encodes more than the fusion algebra of fractionalized quasiparticles. We apply our theory to Kitaev's toric code and Levin-Wen string-net models. We predict that the Z2{Z}_{2}Z2​ toric code and Z2{Z}_{2}Z2​ double-semion model [more generally, the Zk{Z}_{k}Zk​ gauge theory and the $U{(1)}_{k}\ifmmode\times\else\texttimes\fi{}U{(1)}_{\ensuremath{-}k}$ nonchiral fractional quantum Hall state at even integer kkk] can be numerically and experimentally distinguished, by measuring their boundary degeneracy on an annulus or a cylinder.
Abstract One of the most striking features of gapped quantum phases that exhibit topological order is the presence of long-range entanglement that cannot be detected by any local order parameter. The formalism of projected entangled-pair states is a natural framework for the parameterization of gapped ground state wavefunctions which allows one to characterize topological order in terms of the virtual symmetries of the local tensors that encode the wavefunction. In their most general form, these symmetries are represented by matrix product operators acting on the virtual level, which leads to a set of algebraic rules characterizing states with topological quantum order. This construction generalizes the concepts of G{\mathsf {G}}G G - and twisted injectivity; the corresponding matrix product operators encode all topological features of the theory and provide a complete picture of the ground state manifold on the torus. We show how the string-net models of Levin and Wen fit within this formalism and in doing so provide a particularly intuitive interpretation of the pentagon equation for F-symbols as the pulling of matrix product operators through the string-net tensor network. Our approach paves the way to finding novel topological phases beyond string nets and elucidates the description of topological phases in terms of entanglement Hamiltonians and edge theories.