标题： 具有伊辛融合范畴对称性的量子簇态自旋链：弱霍普夫对称性TFT的视角 显示英文标题

标题： Quantum Cluster State Spin Chain with Ising Fusion Category Symmetry: A Perspective from Weak Hopf SymTFT

摘要： 在这项工作中，我们提出了一种簇态晶格哈密顿量的构造，该构造表现出伊辛融合代数的对称性。该构造是在弱霍普夫对称拓扑场理论（SymTFT）的框架内提出的，我们为弱霍普夫量子双代数模型分配了平滑和粗糙边界，从而扩展了传统的簇态模型。我们构造的核心是弱霍普夫伊辛边界管代数$\mathcal{T}_{\mathsf{Ising}}$，其表示范畴等价于伊辛融合范畴$\mathsf{Ising}$。我们将这个代数作为弱霍普夫量子双代数模型的输入数据。该模型在开和闭的$1\text{d}$流形上均表现出伊辛融合对称性。在开流形上，对称性由$\mathcal{T}_{\mathsf{Ising}} \otimes \mathcal{T}_{\mathsf{Ising}}^{\vee}$控制；在闭流形上，它简化为$\operatorname{Cocom}(\mathcal{T}_{\mathsf{Ising}}) \otimes \operatorname{Cocom}(\mathcal{T}_{\mathsf{Ising}}^{\vee})$。由于伊辛融合代数嵌入到$\operatorname{Cocom}(\mathcal{T}_{\mathsf{Ising}}^{\vee})$中，该模型忠实实现了伊辛融合范畴的对称性。 显示英文摘要

摘要： In this work, we present a construction of a cluster state lattice Hamiltonian that exhibits the symmetry of the Ising fusion algebra. This construction is formulated within the framework of weak Hopf symmetry topological field theory (SymTFT), where we assign smooth and rough boundaries to the weak Hopf quantum double model, thereby extending the conventional cluster state model. Central to our construction is the weak Hopf Ising boundary tube algebra $\mathcal{T}_{\mathsf{Ising}}$, whose representation category is equivalent to the Ising fusion category $\mathsf{Ising}$. We take this algebra as the input data for the weak Hopf quantum double model. The resulting model exhibits Ising fusion symmetry on both open and closed $1\text{d}$ manifolds. On open manifolds, the symmetry is governed by $\mathcal{T}_{\mathsf{Ising}} \otimes \mathcal{T}_{\mathsf{Ising}}^{\vee}$; on closed manifolds, it reduces to $\operatorname{Cocom}(\mathcal{T}_{\mathsf{Ising}}) \otimes \operatorname{Cocom}(\mathcal{T}_{\mathsf{Ising}}^{\vee})$. Since the Ising fusion algebra embeds into $\operatorname{Cocom}(\mathcal{T}_{\mathsf{Ising}}^{\vee})$, the model faithfully realizes the symmetry of the Ising fusion category.