$N$-ality symmetry and SPT phases in (1+1)d
Jun Maeda, Tsubasa Oishi
TL;DR
The paper addresses non-invertible $N$-ality symmetries in (1+1)D by realizing them through gauging a non-anomalous subgroup of $\mathbb{Z}_N\times\mathbb{Z}_N\times\mathbb{Z}_N$ with a type III anomaly. It develops two complementary methods to compute fusion rules: a direct field-theoretic construction and a representation-theoretic viewpoint, and then uses the SymTFT framework to classify symmetry-protected topological (SPT) phases associated with the $N$-ality symmetry, with precise counting formulas depending on the parity of $N$. The $N$-ality category is shown to be equivalent to $\mathrm{Rep}(G)$ for a Heisenberg-type group $G=(\mathbb{Z}_N^{A}\times\mathbb{Z}_N^{B})\rtimes_{\rho}\mathbb{Z}_N^{C}$, with defects $\mathcal N_k$ mapped to irreducible representations; fusion rules follow from character theory. The authors construct explicit lattice Hamiltonians realizing several non-invertible SPT phases and discuss interfaces that relate these phases to conventional $\mathbb{Z}_N^{e}\times\mathbb{Z}_N^{o}$ SPTs, thereby linking categorical data to concrete lattice models. Overall, the work provides a thorough classification of non-invertible SPTs for $N$-ality symmetry and concrete lattice realizations, illuminating the structure of non-invertible symmetries in low dimensions and their physical consequences.
Abstract
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.