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

TL;DR

This work develops a comprehensive framework for (1+1)D phases with non-invertible, fusion-category symmetries by elevating weak Hopf algebras as the underlying symmetry algebra. Through a symmetry-topological-field-theory (SymTFT) lens, the authors realize arbitrary fusion-category symmetries on a lattice via a two-boundary (smooth and rough) cluster ladder built from a weak Hopf gauge theory, with symmetry living in $H \times \hat{H}$ and reducing to cocommutative subalgebras on closed manifolds. They introduce weak Hopf tensor-network methods to obtain exact solutions and demonstrate that boundary data (via comodule algebras and tube algebras) reconstruct the weak Hopf symmetry from a given fusion category. The framework unifies Abelian and non-Abelian cases, provides explicit constructions (including $H_8$ and Fibonacci examples), and expresses the symmetry as dual Hopf data, paving the way for realizing and classifying (1+1)D phases with categorical symmetries and exploring potential quantum-information applications.

Abstract

We introduce weak Hopf symmetry as a tool to explore (1+1)-dimensional topological phases with non-invertible symmetries. Drawing inspiration from Symmetry Topological Field Theory (SymTFT), we construct a lattice model featuring two boundary conditions: one that encodes topological symmetry and another that governs non-topological dynamics. This cluster ladder model generalizes the well-known cluster state model. We demonstrate that the model exhibits weak Hopf symmetry, incorporating both the weak Hopf algebra and its dual. On a closed manifold, the symmetry reduces to cocommutative subalgebras of the weak Hopf algebra. Additionally, we introduce weak Hopf tensor network states to provide an exact solution for the model. As every fusion category corresponds to the representation category of some weak Hopf algebra, fusion category symmetry naturally corresponds to a subalgebra of the dual weak Hopf algebra. Consequently,the cluster ladder model offers a lattice realization of arbitrary fusion category symmetries.

Figures (4)

• Figure 1: An illustration of a quantum double lattice, viewed as a cellulation of a surface. The purple region indicates a ribbon on the lattice.
• Figure 2: Depiction of the SymTFT sandwich: the symmetry boundary is drawn in cyan, while the physical boundary is drawn in magenta.
• Figure 3: Correspondence between the cluster state model and the quantum double model with one smooth boundary and one rough boundary. (a) The black lattice denotes the quantum double lattice, where physical degrees of freedom reside on solid edges. In the corresponding cluster state model, qubits are placed on vertices, with odd vertices shown in red, even vertices in blue, and links between them in gray. (b) The magnetic $X$-string operator on the rough boundary corresponds to the $\operatorname{Cocom}(H)$ symmetry operator of the cluster state model. (c) The electric $Z$-string operator on the smooth boundary corresponds to the $\operatorname{Cocom}(\hat{H})$ symmetry operator of the cluster state model.
• Figure 4: Tensor network representation of weak Hopf cluster states. The bottom layer represents the cluster lattice, where red circles denote odd vertices and blue rectangles denote even vertices. The tensor network above the lattice is expressed in terms of the structure constants of the weak Hopf algebra.

Theorems & Definitions (25)

• Definition 1: Hopf symmetry
• Definition 2
• Definition 3: Weak Hopf symmetry
• Definition 4
• Remark 3.1
• Lemma 1: Schur's lemma for weak Hopf algebra
• proof
• Lemma 2
• proof
• Theorem 1