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

TL;DR

This paper introduces weak Hopf symmetry in (1+1)D topological phases, constructs a lattice model with boundary conditions encoding these symmetries, and provides an exact solution using weak Hopf tensor networks.

Contribution

It develops a lattice model exhibiting weak Hopf symmetry, generalizes the cluster state, and realizes arbitrary fusion category symmetries on a lattice.

Findings

The model exhibits weak Hopf symmetry with both algebra and dual algebra structures.

Weak Hopf tensor network states provide an exact solution for the model.

The lattice realization captures arbitrary fusion category symmetries.

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.