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

TL;DR

The work develops a microscopic framework for 2+1d SPT phases protected by non-invertible fusion 2-category symmetries, realized by gauging a subgroup $H$ of a finite group $G$ to form the group-theoretical symmetry $\\mathcal{C}(G; H)$. It introduces generalized cluster states as exactly solvable lattice realizations, providing explicit tensor-network representations for ground states and symmetry operators, including fractionalized actions on virtual indices. The paper then analyzes interfaces between different SPT phases through the strip 2-algebra, showing interface degeneracy and principled gapped-interface classifications, and develops $S^1$-parameterized families to realize generalized Thouless pumps via self-interface modes. Concrete examples, including $2\mathrm{Rep}(G_0)\boxtimes 2\mathrm{Vec}_{G_0}$ and $2\mathrm{TY}(A,1)$, illustrate the general theory, yielding detailed tensor-network constructions, interface structures, and pumping phenomena. Together, these results advance the understanding of bulk-boundary phenomena, non-invertible symmetries, and parameterized topological transport in higher-dimensional lattice systems.”

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.

Figures (5)

• Figure 1: An object $X$ corresponds to a topological surface. A 1-morphism $x: Y \to X$ corresponds to a topological line between $X$ and $Y$. A 2-morphism $\chi: y \Rightarrow x$ corresponds to a topological point operator between $x$ and $y$. The magenta arrows perpendicular to the surfaces represent their coorientation.
• Figure 2: The fusion of topological surfaces corresponds to the tensor product of objects, while the fusion of topological lines corresponds to the composition of 1-morphisms.
• Figure 3: The square lattice on which our model is defined. We impose periodic boundary conditions in both directions. The orientation of the edges is fixed as shown above.
• Figure 4: A loop $\gamma$ on the dual lattice.
• Figure 5: The Wilson line $W^{(i_0i)}$ is the product of the Pauli-Z operators on the edges intersecting a path between $i_0$ and $i$. This operator does not depend on the choice of a path because we imposed the flatness condition on the state space.