Generalized cluster states in 2+1d: non-invertible symmetries, interfaces, and parameterized families

TL;DR

This paper constructs 2+1D lattice models of SPT phases with non-invertible symmetries, exploring their interfaces, degeneracies, and topological charge pumping, thus extending the understanding of symmetry-protected topological phases.

Contribution

It introduces generalized cluster models with non-invertible symmetries, analyzes their interfaces, and demonstrates parameterized families exhibiting topological charge pumping phenomena.

Findings

Interfaces between different SPT phases are degenerate.

Constructed parameterized families show generalized Thouless pump.

Models exhibit non-invertible symmetries described by fusion 2-categories.

Abstract

We construct 2+1-dimensional lattice models of symmetry-protected topological (SPT) phases with non-invertible symmetries and investigate their properties using tensor networks. These models, which we refer to as generalized cluster models, are constructed by gauging a subgroup symmetry $H \subset G$ in models with a finite group 0-form symmetry $G$. By construction, these models have a non-invertible symmetry described by the group-theoretical fusion 2-category $\mathcal{C}(G; H)$. After identifying the tensor network representations of the symmetry operators, we study the symmetry acting on the interface between two generalized cluster states. In particular, we will see that the symmetry at the interface is described by a multifusion category known as the strip 2-algebra. By studying possible interface modes allowed by this symmetry, we show that the interface between generalized cluster states in different SPT phases must be degenerate. This result generalizes the ordinary bulk-boundary correspondence. Furthermore, we construct parameterized families of generalized cluster states and study the topological charge pumping phenomena, known as the generalized Thouless pump. We exemplify our construction with several concrete cases, and compare them with known phases, such as SPT phases with $2\mathrm{Rep}((\mathbb{Z}_{2}^{[1]}\times\mathbb{Z}_{2}^{[1]})\rtimes\mathbb{Z}_{2}^{[0]})$ symmetry.