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

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.

33 pages, 2 figures; v2: references added; v3: a minor correction to the classification of SPT phases

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• Non-invertible translation from Lieb-Schultz-Mattis anomaly