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<p>We propose weak Hopf symmetry as a general framework to explore (1+1)D topological phases that exhibit non-invertible symmetries. Inspired by the Symmetry Topological Field Theory (SymTFT) description of quantum phases with non-invertible symmetry, we construct a lattice model by introducing two distinct topological boundary conditions for a weak Hopf lattice gauge theory. One boundary encodes the topological symmetry information, while the other incorporates the non-topological dynamics. The resulting model is termed the cluster ladder model. We demonstrate that the cluster state model is a special case of this broader class of lattice models exhibiting weak Hopf symmetry <i>H</i> × <i>Ĥ</i>, where <i>H</i> is a weak Hopf algebra and <i>Ĥ</i> is its dual weak Hopf algebra. On a closed manifold, the symmetry reduces to Cocom(<i>H</i>) × Cocom(<i>Ĥ</i>), corresponding to the cocommutative subalgebras of <i>H</i> × <i>Ĥ</i>. An essential weak Hopf sub-symmetry is Cocom(<i>H</i>) × Rep(<i>H</i>), which, in the finite group case, reduces to the familiar symmetry <i>G</i> × Rep(<i>G</i>). To exactly solve the lattice model, we introduce a weak Hopf tensor network. Furthermore, we demonstrate how to construct the lattice realization of an arbitrary fusion category symmetry <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">S</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{S} \)</EquationSource> </InlineEquation> via combining Tannaka-Krein reconstruction or weak Hopf tube algebra and the cluster ladder model.</p>

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Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory

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  • 发表日期：

    2026年3月16日

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  • Zhian Jia

摘要

We propose weak Hopf symmetry as a general framework to explore (1+1)D topological phases that exhibit non-invertible symmetries. Inspired by the Symmetry Topological Field Theory (SymTFT) description of quantum phases with non-invertible symmetry, we construct a lattice model by introducing two distinct topological boundary conditions for a weak Hopf lattice gauge theory. One boundary encodes the topological symmetry information, while the other incorporates the non-topological dynamics. The resulting model is termed the cluster ladder model. We demonstrate that the cluster state model is a special case of this broader class of lattice models exhibiting weak Hopf symmetry H × Ĥ, where H is a weak Hopf algebra and Ĥ is its dual weak Hopf algebra. On a closed manifold, the symmetry reduces to Cocom(H) × Cocom(Ĥ), corresponding to the cocommutative subalgebras of H × Ĥ. An essential weak Hopf sub-symmetry is Cocom(H) × Rep(H), which, in the finite group case, reduces to the familiar symmetry G × Rep(G). To exactly solve the lattice model, we introduce a weak Hopf tensor network. Furthermore, we demonstrate how to construct the lattice realization of an arbitrary fusion category symmetry S \( \mathcal{S} \) via combining Tannaka-Krein reconstruction or weak Hopf tube algebra and the cluster ladder model.