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Local topological order and boundary algebras

Corey Jones, Pieter Naaijkens, David Penneys, Daniel Wallick

TL;DR

This work axiomatizes local topological order for quantum spin systems through nets of local ground-state projections and constructs a boundary net that holographically encodes bulk topological order without a boundary Hamiltonian. By proving that Toric Code and Levin–Wen models satisfy the LTO axioms, it establishes a canonical bulk–boundary correspondence via a boundary algebra net, and analyzes boundary states, KMS structure, and cone algebras. A central contribution is the identification of the bulk topological order with the braided DHR bimodule category of the boundary net (DHR(B) ≅ Z(C)) in 2+1 D, yielding a rigorous, algebraic formulation of topological holography for these models. The paper also clarifies how cone algebra types (II∞ vs III) depend on the pointedness of the underlying fusion category and develops a detailed program to extract bulk information from boundary data through DHR theory and tube/skein structures. Overall, it provides a Hamiltonian-free, operator-algebraic framework for bulk–boundary duality in topologically ordered phases, with precise categorial characterizations of bulk excitations and boundary states.

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 [Ann. H. Poincaré 25, 2024] that the bulk cone von Neumann algebra in the Toric Code is of type $\rm{II}$, and we show that Levin-Wen models can have cone algebras of type $\rm{III}$. 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.

Local topological order and boundary algebras

TL;DR

This work axiomatizes local topological order for quantum spin systems through nets of local ground-state projections and constructs a boundary net that holographically encodes bulk topological order without a boundary Hamiltonian. By proving that Toric Code and Levin–Wen models satisfy the LTO axioms, it establishes a canonical bulk–boundary correspondence via a boundary algebra net, and analyzes boundary states, KMS structure, and cone algebras. A central contribution is the identification of the bulk topological order with the braided DHR bimodule category of the boundary net (DHR(B) ≅ Z(C)) in 2+1 D, yielding a rigorous, algebraic formulation of topological holography for these models. The paper also clarifies how cone algebra types (II∞ vs III) depend on the pointedness of the underlying fusion category and develops a detailed program to extract bulk information from boundary data through DHR theory and tube/skein structures. Overall, it provides a Hamiltonian-free, operator-algebraic framework for bulk–boundary duality in topologically ordered phases, with precise categorial characterizations of bulk excitations and boundary states.

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 , we define a local net of boundary algebras on , 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 [Ann. H. Poincaré 25, 2024] that the bulk cone von Neumann algebra in the Toric Code is of type , and we show that Levin-Wen models can have cone algebras of type . 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.
Paper Structure (23 sections, 41 theorems, 161 equations, 1 algorithm)

This paper contains 23 sections, 41 theorems, 161 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Nets of algebras and local topological order
  3. Nets of algebras and nets of projections
  4. Local topological order axioms
  5. Canonical state of a locally topologically ordered net of projections
  6. The boundary net
  7. Example: Kitaev's Toric Code
  8. Example: Levin-Wen string nets
  9. Boundary states and applications to cone algebras
  10. Toric Code boundary states
  11. The canonical Levin-Wen string net boundary state
  12. The KMS condition for the Levin-Wen string net boundary state
  13. Cone algebras
  14. Additional boundary states on the Levin-Wen boundary net
  15. Bulk topological order from the boundary algebra net in 2+1 D
  16. ...and 8 more sections

Key Result

Theorem 1

The Toric Code and Levin-Wen models satisfy the LTO axioms LTO:Hastings--LTO:Injective. The boundary nets are fusion categorical nets over the lattice ${\mathbb Z}$.

Theorems & Definitions (116)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Definition 2.1: Nets of algebras
  • Remark 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6: Bounded spread isomorphism
  • ...and 106 more