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arXiv:2509.23734

Haag Duality for 2D Quantum Spin Systems

Yoshiko Ogata, David Pérez-Garcı́a, Alberto Ruiz-de-Alarcón

Published Sep 28, 2025• 0 citations

Abstract

Haag duality is a fundamental locality property introduced in the pioneering formulation of algebraic quantum field theory by Haag and Kastler in the 1960s. Since then, it has played a central role, most notably in the classification of superselection sectors by Doplicher, Haag, and Roberts in the 1970s. Over the past two decades, this concept has migrated from its relativistic origins to quantum spin systems, becoming a cornerstone of the operator-algebraic approach to the long-standing problem of classifying two-dimensional topological quantum phases of matter. In physics, it is widely conjectured that such phases are classified by their emergent anyons, a view supported by exactly solvable models exemplifying all known non-chiral phases: Kitaev's quantum double models, Levin-Wen string-net models, and their slight generalizations. In these models, elementary excitations behave as quasi-particles, namely anyons, whose fusion and braiding properties form a tensor category expected to characterize the phase of matter. A major open problem was to derive the emergence of anyons and the stability of their fusion and braiding beyond these solvable models. Recently, it has been shown that a weaker, phase-stable form of Haag duality resolves these questions. However, rigorous proofs of Haag duality in two dimensions were previously restricted to systems exhibiting abelian anyons. In this work, we establish Haag duality for a broad class of tensor network models based on $C^*$-weak Hopf algebras, encompassing all Kitaev quantum double and Levin-Wen string-net models, and expected to include all non-chiral topological quantum phases of matter.

DOI: 10.48550/arxiv.2509.23734

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Haag duality is a fundamental locality property introduced in the pioneering formulation of algebraic quantum field theory by Haag and Kastler in the 1960s. Since then, it has played a central role, most notably in the classification of superselection sectors by Doplicher, Haag, and Roberts in the 1970s. Over the past two decades, this concept has migrated from its relativistic origins to quantum spin systems, becoming a cornerstone of the operator-algebraic approach to the long-standing problem of classifying two-dimensional topological quantum phases of matter. In physics, it is widely conjectured that such phases are classified by their emergent anyons, a view supported by exactly solvable models exemplifying all known non-chiral phases: Kitaev's quantum double models, Levin-Wen string-net models, and their slight generalizations. In these models, elementary excitations behave as quasi-particles, namely anyons, whose fusion and braiding properties form a tensor category expected to characterize the phase of matter. A major open problem was to derive the emergence of anyons and the stability of their fusion and braiding beyond these solvable models. Recently, it has been shown that a weaker, phase-stable form of Haag duality resolves these questions. However, rigorous proofs of Haag duality in two dimensions were previously restricted to systems exhibiting abelian anyons. In this work, we establish Haag duality for a broad class of tensor network models based on

C^*

-weak Hopf algebras, encompassing all Kitaev quantum double and Levin-Wen string-net models, and expected to include all non-chiral topological quantum phases of matter.

Abstract We study Kitaev’s quantum double model for arbitrary finite gauge group in infinite volume, using an operator-algebraic approach. The quantum double model hosts anyonic excitations which can be identified with equivalence classes of ‘localized and transportable endomorphisms’, which produce anyonic excitations from the ground state. Following the Doplicher–Haag–Roberts (DHR) sector theory from AQFT, we organize these endomorphisms into a braided monoidal category capturing the fusion and braiding properties of the anyons. We show that this category is equivalent to

\textbf{Rep}_f {{\mathcal {D}}}(G)

Rep f D ( G ) , the representation category of the quantum double of G . This establishes for the first time the full DHR structure for a class of 2d quantum lattice models with non-abelian anyons.

We establish a direct connection between the entanglement bootstrap and the Doplicher--Haag--Roberts theory of superselection sectors for two-dimensional quantum spin systems. Under the entanglement-bootstrap assumptions, we prove Haag duality and establish a one-to-one correspondence between simple entanglement-bootstrap anyon types and irreducible superselection sectors. For each fusion channel, we further construct an affine bijection from density matrices on the entanglement-bootstrap fusion space to normalized positive intertwiners in the corresponding operator-algebraic fusion channel. These results connect the entropic and operator-algebraic descriptions of anyons directly, without passing through exactly solvable lattice models.

We propose a discrete model---the twisted quantum double model---of 2D topological phases based on a finite group

G

and a 3-cocycle $\ensuremath{\alpha}$ over

G

. The detailed properties of the ground states are studied, and we find that the ground-state subspace can be characterized in terms of the twisted quantum double ${D}^{\ensuremath{\alpha}}(G)$ of

G

. When $\ensuremath{\alpha}$ is the trivial 3-cocycle, the model becomes Kitaev's quantum double model based on the finite group

G

, in which the elementary excitations are known to be classified by the quantum double

D(G)

of

G

. Our model can be viewed as a Hamiltonian extension of the Dijkgraaf-Witten topological gauge theories to the discrete graph case with gauge group being a finite group. We also demonstrate a duality between a large class of Levin-Wen string-net models and certain twisted quantum double models, by mapping the string-net

6j

symbols to the corresponding 3-cocycles.

We prove Haag duality for cone-like regions in the ground state representation corresponding to the translational invariant ground state of Kitaev’s quantum double model for finite abelian groups. This property says that if an observable commutes with all observables localized outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localized in disjoint regions commute. As an application, we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher–Haag–Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double.

Abstract Dualities play a central role in the study of quantum spin chains, providing insight into the structure of quantum phase diagrams and phase transitions. In this work, we study categorical dualities, which are defined as bounded-spread isomorphisms between algebras of symmetry-respecting local operators on a spin chain. We consider generalized global symmetries that correspond to unitary fusion categories, which are represented by matrix-product operator algebras. A fundamental question about dualities is whether they can be extended to quantum cellular automata on the larger algebra generated by all local operators in the the unit matrix-product operator sector. For on-site representations of Hopf algebra symmetries, this larger algebra is the usual tensor product quasi-local algebra. We present a solution to the extension problem using the machinery of Doplicher–Haag–Roberts bimodules. Our solution provides a crisp categorical criterion for when an extension of a duality exists. We show that the set of possible extensions form a torsor over the invertible objects in the relevant symmetry category. As a corollary, we obtain a classification result concerning dualities in the group case.

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.