Generating Lattice Non-invertible Symmetries

TL;DR

This paper develops methods to generate new lattice non-invertible symmetries from existing ones, revealing novel symmetries like the dipole Kramers-Wannier symmetry and analyzing their topological and anomaly properties.

Contribution

It introduces a systematic approach to constructing new lattice non-invertible symmetries and identifies a novel dipole Kramers-Wannier symmetry with detailed topological analysis.

Findings

New non-invertible symmetry in models with $ ext Z_N$ dipole symmetries.

Construction of symmetries via composition and sandwiching transformations.

Analysis of the topological defect, anomaly, and generalized transformations of the new symmetry.

Abstract

Lattice non-invertible symmetries have rich fusion structures and play important roles in understanding various exotic topological phases. In this paper, we explore methods to generate new lattice non-invertible transformations/symmetries from a given non-invertible seed transformation/symmetry. The new lattice non-invertible symmetry is constructed by composing the seed transformations on different sites or sandwiching a unitary transformation between the transformations on the same sites. In addition to known non-invertible symmetries with fusion algebras of Tambara-Yamagami $\mathbb Z_N\times\mathbb Z_N$ type, we obtain a new non-invertible symmetry in models with $\mathbb Z_N$ dipole symmetries. We name the latter the dipole Kramers-Wannier symmetry because it arises from gauging the dipole symmetry. We further study the dipole Kramers-Wannier symmetry in depth, including its topological defect, its anomaly and its associated generalized Kennedy-Tasaki transformation.