$N$-ality symmetry and SPT phases in (1+1)d

Duality symmetries have been extensively investigated in various contexts, playing a crucial role in understanding quantum field theory and condensed matter theory. In this paper, we extend this framework by studying $N$-ality symmetries, which are a generalization of duality symmetries and are mathematically described by $\mathbb{Z}_N$-graded fusion categories. In particular, we focus on an $N$-ality symmetry obtained by gauging a non-anomalous subgroup of $\mathbb{Z}_N\times\mathbb{Z}_N\times\mathbb{Z}_N$ symmetry with a type III anomaly. We determine the corresponding fusion rules via two complementary approaches: a direct calculation and a representation-theoretic method. As an application, we study the symmetry-protected topological (SPT) phases associated with the $N$-ality symmetry. We classify all such SPT phases using the SymTFT framework and explicitly construct lattice Hamiltonians for some of them.

💡 Research Summary

This paper presents a comprehensive study of non-invertible symmetries and symmetry-protected topological (SPT) phases in (1+1) dimensions. The central focus is the generalization of well-known duality symmetries (like Kramers-Wannier duality in the Ising model) to so-called “$N$-ality symmetries.” These are described mathematically by $\mathbb{Z}_N$-graded fusion categories.

The authors construct a specific $N$-ality symmetry by starting with a $\mathbb{Z}_N \times \mathbb{Z}_N \times \mathbb{Z}_N$ global symmetry that possesses a Type III mixed anomaly, characterized by the topological action $ \frac{2\pi i}{N} \int a \cup b \cup c $. They then gauge a non-anomalous subgroup, $\mathbb{Z}_N \times \mathbb{Z}_N$. This gauging process transmutes the original invertible symmetries into a non-invertible symmetry structure. The resulting symmetry category is the $N$-ality symmetry.

A major technical achievement of the paper is the computation of the fusion rules for the non-invertible defects in this $N$-ality category. The authors employ two complementary approaches. The first is a direct field-theoretic calculation, where defects are expressed as functionals of gauge fields coupled to dynamical fields on their worldlines. The second, and particularly powerful method, is a representation-theoretic approach. The paper shows that the $N$-ality symmetry category in question is equivalent to the representation category $\text{Rep}(G)$ of a certain non-Abelian group $G$. This allows the defects to be identified with characters of irreducible representations of $G$, enabling a systematic derivation of fusion rules using group representation theory, which works efficiently even for composite $N$.

As a key application, the paper investigates SPT phases protected by this non-invertible $N$-ality symmetry. Utilizing the Symmetry Topological Field Theory (SymTFT) framework, the authors establish a one-to-one correspondence between gapped phases of the original anomalous $\mathbb{Z}_N^3$ theory and gapped phases of the gauged theory with $N$-ality symmetry. This correspondence is used to classify all possible SPT phases under the $N$-ality symmetry. The classification formula depends on whether $N$ is odd or even, counting the distinct phases. To ground the theoretical analysis, the authors explicitly construct lattice Hamiltonian models realizing some of these SPT phases. Furthermore, by constructing an interface Hamiltonian between two such models, they demonstrate that these non-invertible SPT phases can be in the same phase as SPT phases protected by a conventional invertible sub-symmetry ($\mathbb{Z}_N \times \mathbb{Z}_N$).

In summary, the paper makes significant contributions by providing a concrete mechanism for generating $N$-ality symmetries, offering powerful tools (especially the Rep(G) isomorphism) for calculating their algebraic data, and delivering a complete classification and explicit models for their associated SPT phases, thereby advancing the understanding of non-invertible symmetries in condensed matter and quantum field theory.