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arXiv:2604.00910

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

Hrant Topchyan

Published Apr 1, 2026• 0 citations
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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 the 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, directly realizing the corresponding 't Hooft anomaly.

DOI
10.48550/arxiv.2604.00910

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Two-dimensional topological paramagnets protected by Z3\mathbb{Z}_3Z3​ symmetry: Properties of the boundary Hamiltonian
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It is well known that symmetry-protected topological (SPT) phases host nontrivial boundaries that cannot be mimicked in a lower-dimensional system with a conventional realization of symmetry. However, for SPT phases of bosons (fermions) within the cohomology (supercohomology) classification the boundary can be recreated without the bulk at the cost of a non-onsite-symmetry action. This raises the following question: Can one also mimic the boundaries of SPT phases which lie outside the (super)cohomology classification? In this paper, we study this question in the context of (2+1)(2+1)(2+1)-dimensional [(2+1)D][(2+1)\mathrm{D}][(2+1)D] fermion SPTs. We focus on the root SPT phase for the symmetry group $G={Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$. Starting with an exactly solvable model for the bulk of this phase constructed by Tarantino and Fidkowski, we derive an effective one-dimensional (1D) lattice model for the boundary. Crucially, the Hilbert space of this 1D model does not have a local tensor product structure, but rather is obtained by placing a local constraint on a local tensor product Hilbert space. We derive the action of the Z2{Z}_{2}Z2​ symmetry on this Hilbert space and find a simple three-site Hamiltonian that respects this symmetry. We study this Hamiltonian numerically using exact diagonalization and DMRG and find strong evidence that it realizes an Ising conformal field theory where the Z2{Z}_{2}Z2​ symmetry acts as the Kramers-Wannier duality; this is the expected stable gapless boundary state of the present SPT. A simple modification of our construction realizes the boundary of the (2+1)D(2+1)\mathrm{D}(2+1)D topological superconductor protected by time-reversal symmetry T\mathcal{T}T with ${\mathcal{T}}^{2}={(\ensuremath{-}1)}^{\mathcal{F}}$.
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