Non-invertible SPTs: an on-site realization of (1+1)d anomaly-free fusion category symmetry

TL;DR

This work develops an onsite, lattice-based framework for (1+1)d SPTs protected by fusion category symmetries, showing that anomaly-free realizations require a fiber functor and are captured by Q-systems in the charge category. It provides an explicit Rep$^ abla(D_8)$ construction realizing three SPT phases related by an $S_3$-duality, and shows how Hopf C$^*$-algebras underpin a microscopic MPO realization of the symmetry. Ground-state and edge phenomena are characterized via the Q-system, its forgetful image, and the associated fixed-point algebras, linking bulk topology to robust edge modes. The categorical framework, including Morita equivalence and Tannaka duality, clarifies how different fiber functors map to distinct or equivalent SPT phases and how defect fusion encodes phase structure, with implications for general non-invertible symmetries in higher dimensions and fermionic settings.

Abstract

We investigate (1+1)d symmetry-protected topological (SPT) phases with fusion category symmetries. We emphasize that the UV description of an anomaly-free fusion category symmetry must include the fiber functor, giving rise to a local symmetry action, a charge category and a trivial phase. We construct an ``onsite'' matrix-product-operator (MPO) version of the Hopf algebra symmetry operators in a lattice model with tensor-product Hilbert space. In particular, we propose a systematic framework for classifying and constructing SPTs with non-invertible symmetries. An SPT phase corresponds to a Q-system in the charge category, such that the Q-system becomes a matrix algebra when the symmetry is forgotten. As an example, we provide an explicit microscopic realization of all three $\mathsf{Rep}^\dagger(D_8)$ SPT phases, including a trivial phase, and further demonstrate the $S_3$-duality among these three SPT phases.

Figures (5)

• Figure 1: A $\mathsf{Rep}^\dagger(D_8)$-symmetric chain, where black dots are lattice sites indexed by integers. On each site $i\in\mathbb{Z}$, there is a $\mathsf{Rep}^\dagger(D_8)$-charge, i.e. a $D_8$-graded Hilbert space $V_i$. The total Hilbert space is the tensor product of $D_8$-graded Hilbert spaces at each site respecting the ordering of lattice sites.
• Figure 2: A TDL is provided by an MPO in the continuum limit. Each crossing at the right-hand side represents a local tensor of the tensor network representation of the partition function.
• Figure 3: Illustration of the categorical relationship of SPT phases. ${\cal C}$ represents the symmetry category, ${\cal C} _ {\cal M} ^\vee$ represents the charge category, and $\mathsf{Fun}_ {\cal C} ( {\cal M} ,{}_{\pi} {\cal M} )$ represents the category of projective charges. Left ${\cal C}$ module ${\cal M}$ describes the lattice construction. For tensor product Hilbert space, ${\cal M} = \mathsf{Hilb}$.
• Figure 4: Illustration of the categorical relationship of SPT phases. Here the pair $( {\cal C} ,f)$ defines a symmetry as well as the trivial phase, and ${\cal C} _{\mathsf{Hilb}_f}^\vee$ the charge category corresponding to this symmetry. The pair $( {\cal C} ,h)$ defines another symmetry, whose charge category is ${\cal C} _{\mathsf{Hilb}_h}^\vee$, which may or may not be the same as ${\cal C} _{\mathsf{Hilb}_f}^\vee$.
• Figure 5: Illustration of the macroscopic local regions. Here the green region represents the left boundary, the blue region represents the bulk and the grey region represents the right boundary.

Theorems & Definitions (63)

• Definition 3.1: C$^\star$-algebra
• Definition 3.2
• Definition 3.3
• Remark 1
• Example 1
• Example 2
• Example 3
• Definition 3.4
• Example 4
• Example 5