The $\mathbb{Z}_N^{\times 3}$ symmetry protected boundary modes in two-dimensional Potts paramagnets

TL;DR

This paper constructs and analyzes boundary Hamiltonians for 2D topological phases with $ ext{Z}_N^{ imes 3}$ symmetry, revealing their algebraic structures and anomalous symmetry representations.

Contribution

It explicitly derives boundary models for $ ext{Z}_N^{ imes 3}$ symmetric phases, highlighting their algebraic structures and anomaly realizations, with a focus on prime and composite N.

Findings

Boundary models are governed by Temperley-Lieb algebras for prime N.

Models exhibit hierarchical structures for composite N.

Global symmetry acts projectively, indicating an 't Hooft anomaly.

Abstract

We construct and analyze a class of one-dimensional boundary Hamiltonians arising from two-dimensional symmetry-protected topological phases with $\mathbb{Z}_N^{\times 3}$ symmetry on a triangular lattice. Using a cohomology-based transformation, the lattice models for the edge modes are explicitly obtained, and their structure is shown to be governed by the arithmetic properties of $N$. For prime $N$, the boundary theory admits a formulation in terms of mutually commuting Temperley-Lieb algebras. For the composite values of $N$, the models exhibit hierarchical or factorized structures. We demonstrate that all phases can be understood in terms of primary models augmented by local defect degrees of freedom that partition the system into independent segments. Finally, the global symmetry is realized on the boundary in a non-on-site and anomalous manner via a projective representation, directly realizing the corresponding 't Hooft anomaly.