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arXiv:2106.15741 · Journal of Mathematical Physics

A derivation of braided C*-tensor categories from gapped ground states satisfying the approximate Haag duality

Yoshiko Ogata

Published Jan 1, 2022• 22 citations
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Abstract

We derive braided C*-tensor categories from gapped ground states on two-dimensional quantum spin systems satisfying some additional condition, which we call the approximate Haag duality.
DOI
10.1063/5.0061785

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We introduce a natural mathematical definition of boundary states of a bulk gapped ground state, in the operator algebraic framework of 222-d quantum spin systems. With approximate Haag duality at the boundary, we derive a C∗C^*C∗-tensor category M~\tilde{\mathcal{M}}M~ out of such boundary state. Under a non-triviality condition of the braiding in the bulk, we show that the Drinfeld center (with an asymptotic constraint) of M~\tilde{\mathcal{M}}M~ is equivalent to the bulk braided C∗C^*C∗-tensor category derived in [14].
The classification of mixed-state topological order requires indices that behave monotonically under finite-depth quantum channels. In two dimensions, a braided C∗C^*C∗-tensor category, which corresponds to strong symmetry, arises from a state satisfying approximate Haag duality. In this note, we show that the SSS-matrix and topological twists of the braided C∗C^*C∗-tensor category are quantities that are monotone under finite-depth quantum channels.
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∗C^*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.
The article arXiv:2410.21454 constructed a braided W∗\mathrm{W^*}W∗-tensor category of superselection sectors associated to a net of von Neumann algebras on a suitably geometric poset using localized and transportable endomorphisms. In this article, we construct an equivalent tensor product and braiding using the Connes fusion tensor product without localizing, using ideas from arXiv:1812.04470 for conformal nets. As an application, we prove that the category of superselection sectors associated to the 2D braided quantum spin system constructed from a unitary braided fusion category B\mathcal{B}B in arXiv:2506.19969 is equivalent to Hilb(B)op\mathsf{Hilb}(\mathcal{B})^{\rm op}Hilb(B)op, the opposite of the unitary ind-completion of B\mathcal{B}B, as a braided W∗\mathrm{W^*}W∗-tensor category.
We show that a superselction sector describing a single nonabelian anyon violates Haag duality, which is equivalent to the fundamental quantum-information theoretic principle of uniqueness of purifications. We provide a simple physics argument as well as a rigorous proof using sector theory, and instantiate our result concretely in Levin-Wen models. We also show that the associated ground state does not allow for quantum steering despite being a pure state. Finally, we show that this ground state violates approximate Haag duality, disproving the conjecture that all gapped ground states satisfy this condition. Our result implies that there are gapped phases of matter where Haag duality fails at every point in the phase.