We study algebraic locality principles on a 2+1D closed lattice in the presence of a Gauss law for a non-invertible symmetry. Prior work in arXiv:2509.03589 showed that when enforcing the Gauss law of an invertible symmetry, the principle of "Haag duality" is preserved exactly, and "disjoint additivity" is preserved after appropriate treatment of discreteness artifacts. Here we show that for a large class of non-invertible on-site symmetries, Haag duality is preserved exactly only for sufficiently nice, "cuspless" regions. For cusped regions, we instead have a weak form of Haag duality that requires adding a collar. Our results apply to double models with a purely magnetic constraint, and to the more general framework of constraints induced by the on-site action of a Hopf algebra. In particular, we treat a class of extended string-net models explicitly. We also demonstrate disjoint additivity for double models based on a group, and a weakened form of disjoint additivity in the setting of a general Hopf algebra.
A bstract The algebraic approach to quantum field theory focuses on the properties of local algebras, whereas the study of (possibly non-invertible) global symmetries emphasizes global aspects of the theory and spacetime. We study connections between these two perspectives by examining how either of two core algebraic properties — “additivity” or “Haag duality” — is violated in a 1+1D CFT or lattice model restricted to the symmetric sector of a general global symmetry. For the Verlinde symmetry of a bosonic diagonal RCFT, we find that additivity is violated whenever the symmetry algebra contains an invertible element, while Haag duality is violated whenever it contains a non-invertible element. We find similar phenomena for the Kramers-Wannier and Rep(D 8 ) non-invertible symmetries on spin chains.
A bstract Quantum systems of physical interest are often local, but there are at least three competing perspectives on how “locality” should be formalized: an algebraic framework, a path-integral framework, and a lattice framework. One puzzle in this competition is that systems with higher-form symmetries, which are perfectly local from the path-integral and lattice perspectives, can violate the algebraic principle of “additivity”. In this paper, we propose a resolution to this puzzle by introducing a weaker locality principle, disjoint additivity, which together with Haag duality should always be satisfied in local quantum systems. As evidence, we give examples in which disjoint additivity is preserved when ordinary additivity is violated; we show that Haag duality and disjoint additivity are satisfied in rather general lattice systems with local symmetry constraints; we give examples of nonlocal theories in which either disjoint additivity or Haag duality is violated; and finally we give examples of systems with nonlocal symmetry constraints in which disjoint additivity is violated, but can be restored by passing to a local “SymTFT” system in one higher dimension.
Under local magnetic Gauss constraints, vertex-disjoint regions can have commuting physical observable algebras that do not generate all observables on their union. We construct such counterexamples for every finite non-Abelian group on regular square tori, retaining the full magnetic physical space and original supports inside a fixed contractible disk. A peripheral projector selects inputs on which a joint transition preserves the constraints, while its loop factor alone fails the universal test over physical exterior completions. For a specified support family around a fixed eight-edge square loop, additivity fails if and only if the group is non-Abelian and each inner face is covered by at least one of its two prescribed peripheral edges. Leaving a face uncovered restores full additivity. For the same loop, additivity also holds with any region at vertex-set graph distance at least two. These results refute a recent group-double additivity proposition.
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.
The Kramers-Wannier transformation of the 1+1d transverse-field Ising model exchanges the paramagnetic and ferromagnetic phases and, at criticality, manifests as a non-invertible symmetry. Extending such self-duality symmetries beyond gauging of abelian groups in tensor-product Hilbert spaces has, however, remained challenging. In this work, we construct a generalized 1+1d Ising model based on a finite-dimensional semisimple Hopf algebra
H that enjoys an anomaly-free non-invertible symmetry
Rep(H). We provide an intuitive diagrammatic formulation of both the Hamiltonian and the symmetry operators using a non-(co)commutative generalization of ZX-calculus built from Hopf-algebraic data. When
H is self-dual, we further construct a generalized Kramers-Wannier duality operator that exchanges the paramagnetic and ferromagnetic phases and becomes a non-invertible symmetry at the self-dual point. This enlarged symmetry mixes with lattice translation and, in the infrared, flows to a weakly integral fusion category given by a
Z2 extension of
Rep(H). Specializing to the Kac-Paljutkin algebra
H8, the smallest self-dual Hopf algebra beyond abelian group algebras, we numerically study the phase diagram and identify four of the six
Rep(H8)-symmetric gapped phases, separated by Ising critical lines and meeting at a multicritical point. We also realize all six
Rep(H8)-symmetric gapped phases on the lattice via the
H-comodule algebra formalism, in agreement with the module-category classification of
Rep(H8). Our results provide a unified Hopf-algebraic framework for non-invertible symmetries, dualities, and the tensor product lattice models that realize them.
We use finite group topological lattice gauge theory, also known as the quantum double model, as a lens to explore a notion of topological order enriched by a non-invertible symmetry. For invertible symmetry enriched topological order, there is an established axiomatisation in terms of a G-crossed braided fusion category. We lay the foundations for a generalisation of this notion. By condensing an arbitrary algebra of charges in a quantum double model, we demonstrate that the category of localised excitations in the resulting theory forms a hypergroup-graded extension of the category of deconfined excitations. For every element in the hypergroup, the associated domain wall acts in a typically non-invertible way on these localised excitations. Both this action and the monoidal structure are compatible with the hypergroup grading. The actual categorical action is encoded in a Hopf monad on the category of localised excitations, and gauging the non-invertible symmetry amounts to computing the category of modules over this Hopf monad. Finally, we outline how this framework naturally extends to theories obtained by condensing algebras in a generic string-net model.