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

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 ${\mathcal{T}}_{\mathsf{I}\mathsf{s}\mathsf{i}\mathsf{n}\mathsf{g}}$, whose representation category is equivalent to the Ising fusion category $\mathsf{I}\mathsf{s}\mathsf{i}\mathsf{n}\mathsf{g}$. 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 ${\mathcal{T}}_{\mathsf{I}\mathsf{s}\mathsf{i}\mathsf{n}\mathsf{g}}\otimes {\mathcal{T}}_{\mathsf{I}\mathsf{s}\mathsf{i}\mathsf{n}\mathsf{g}}^{\vee }$; on closed manifolds, it reduces to $\mathrm{Cocom}({\mathcal{T}}_{\mathsf{I}\mathsf{s}\mathsf{i}\mathsf{n}\mathsf{g}})\otimes \mathrm{Cocom}({\mathcal{T}}_{\mathsf{I}\mathsf{s}\mathsf{i}\mathsf{n}\mathsf{g}}^{\vee })$. Since the Ising fusion algebra embeds into $\mathrm{Cocom}({\mathcal{T}}_{\mathsf{I}\mathsf{s}\mathsf{i}\mathsf{n}\mathsf{g}}^{\vee })$, the model faithfully realizes the symmetry of the Ising fusion category.