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

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.

• The paper demonstrates the explicit construction of 1D boundary Hamiltonians in 2D Potts SPT phases via a cohomology-driven unitary transformation.
• It reveals that the complex boundary theory reduces to primary models with commuting projector structures, including Temperley–Lieb algebra realizations, contingent on the arithmetic properties of N.
• The study elucidates a nontrivial 't Hooft anomaly through projective symmetry actions and defect sectors, offering insights for tunable topological quantum computation.

Boundary Modes in $\mathbb{Z}_N^{\times 3}$ SPT Phases from the 2D Potts Model

Introduction and Context

This paper develops explicit one-dimensional (1D) boundary Hamiltonians for two-dimensional (2D) symmetry-protected topological (SPT) phases protected by $\mathbb{Z}_N^{\times 3}$ symmetry in a triangular lattice realization of the Potts paramagnet. The approach is based on a cohomology-driven unitary transformation, leveraging nontrivial 3-cocycles to precisely construct edge modes and analyze their algebraic and topological features.

The theoretical framework is motivated by the systematic group cohomology classification of bosonic SPT phases, particularly the role of short-range entanglement and nontrivial boundary properties such as gapless edge states and anomalous symmetry actions. A central claim is that, despite the high-dimensional cohomology structure, all nontrivial boundary theories in this class can be reduced to a set of “primary models” with local defect degrees of freedom that split the boundary into independent segments.

Lattice Construction and Cohomology Implementation

The starting point is an $N$-state Potts model on a 2D triangular lattice. It is reformulated via a triangular superlattice, where each supernode aggregates three original sites, thus embedding a local $\mathbb{Z}_N^{\times 3}$ symmetry structure. The physical degrees of freedom at each supernode are indexed by color/flavor, providing the required symmetry sectors for cohomological classification.

Figure 1: Superlattice structure—green, blue, and yellow circles indicate $\mathbb{Z}_N$ nodes, grouped into supernodes with red shading and red lines forming the superlattice.

Within this symmetry group, $$H^3(\mathbb{Z}_N^{\times 3}, U(1)) = \mathbb{Z}_N^{\times7}$$ yields distinct SPT classifications, with the paper focusing on the genuinely three-color, non-reducible sector. The explicit cohomology generator $\omega_3^o$ is constructed as $\varepsilon^{n_1^A n_2^B n_3^C}$, where $\varepsilon = \exp(2\pi i / N)$. The non-exactness and closedness of this cocycle are proved algebraically.

A general symmetrized Hamiltonian transformation is implemented via $U_k$, built from this cocycle, yielding the boundary Hamiltonian:

$\mathbb{Z}_N^{\times 3}$0

with the symmetry implementation ultimately non-on-site and anomalous.

Explicit Structure of the Boundary Hamiltonians

A significant technical result is that the structure of the emergent boundary Hamiltonian depends strongly on the arithmetic nature of $\mathbb{Z}_N^{\times 3}$1:

• Prime $\mathbb{Z}_N^{\times 3}$2: A dramatic algebraic simplification occurs ($\mathbb{Z}_N^{\times 3}$3 for $\mathbb{Z}_N^{\times 3}$4), yielding Hamiltonian terms of the projector form:

$\mathbb{Z}_N^{\times 3}$5

with nontrivial dynamics only when neighboring sites are in the same state. The model admits a two-site representation in terms of commuting projectors, and the algebra naturally organizes into two mutually commuting Temperley–Lieb (TL) algebras.

• Prime Powers $\mathbb{Z}_N^{\times 3}$6: The boundary degrees of freedom decompose into $\mathbb{Z}_N^{\times 3}$7 independent $\mathbb{Z}_N^{\times 3}$8 sectors. The Hamiltonian inherits a tensor product structure, with hierarchical constraints projecting onto increasingly refined subspaces as one moves through the “digits” of the radix-$\mathbb{Z}_N^{\times 3}$9 expansion of the site variables.
• General Composite $N$0: The boundary theory factorizes over the prime decomposition of $N$1, with each factor contributing an independent field. The Hamiltonian for a nontrivial phase $N$2 with $N$3 can be expressed in terms of the primary model for $N$4, plus defect fields that effectively cut the chain into segments.

The paper presents explicit forms for these Hamiltonians, discussing their commuting projector structure, their decomposition into projective sectors, and the appearance of defect degrees of freedom that act as dynamical constraints partitioning the boundary Hilbert space.

Algebraic Structure and Temperley–Lieb Algebra Realization

A salient feature for prime $N$5 is that the Hamiltonian organizes naturally into a staggered sum of two mutually commuting TL algebras. After a basis change, projectors $N$6 and $N$7 satisfy:

$N$8

and mutually commute if $N$9. The Hamiltonian simplifies to:

$\mathbb{Z}_N^{\times 3}$0

This structure is suggestive of links to integrable systems, loop models, and exactly solvable statistical mechanics systems. The explicit emergence of the TL algebra provides natural prospects for identifying critical regimes and for establishing connections to conformal field theories (CFTs) in the continuum limit.

Symmetry Analysis and the 't Hooft Anomaly

The global $\mathbb{Z}_N^{\times 3}$1 symmetry is shown to act on the boundary in a non-on-site, projective, and anomalous fashion. The explicit calculation demonstrates a nontrivial associator:

$\mathbb{Z}_N^{\times 3}$2

with the phase $\mathbb{Z}_N^{\times 3}$3 directly reproducing the underlying 3-cocycle, giving a concrete lattice realization of the corresponding 't Hooft anomaly.

Symmetries, Conserved Quantities, and Topological Sectors

The boundary Hamiltonians possess large permutation symmetry groups—$\mathbb{Z}_N^{\times 3}$4 (or products thereof) acting on different sectors—and support extensive sets of nontrivial conserved quantities (winding- and laterality-like). The counting of independent symmetric and antisymmetric charges follows from the structure of the local degrees of freedom and their interaction constraints.

Figure 2: A geometric depiction of the winding charge $\mathbb{Z}_N^{\times 3}$5 for $\mathbb{Z}_N^{\times 3}$6, corresponding to the number of revolutions of the site configuration around a virtual axis.

Additionally, in the particle-like representation, the number of “active elements” or morphable sites matches the number of parenthesis pairs in a maximally nested description, indicating nontrivial excitation and degeneracy structure.

Defect Sectors and Hierarchical Decomposition

For composite $\mathbb{Z}_N^{\times 3}$7 and non-primary phases, local defects arise (described by projectors $\mathbb{Z}_N^{\times 3}$8 with nontrivial degeneracy), acting as splitting points that partition the chain into independent segments. The low-energy states are defect-free; the existence of defects is energetically penalized.

This hierarchical organization leads to a systematic reduction of all phases to primary models, plus controlled defect decorations, which is argued to capture the full topological and dynamical content of the family.

Implications and Prospects

The explicit boundary Hamiltonians and their algebraic structure clarify the SPT–anomaly correspondence in $\mathbb{Z}_N^{\times 3}$9-protected Potts models and provide an organizing principle for boundary phase diagrams across all $\mathbb{Z}_N$0. The presence of TL algebra connections motivates investigation into integrability, CFT description of the continuum limit, and possible mappings to loop and RSOS models.

Of practical interest, the genericization to composite $\mathbb{Z}_N$1 and the realization of dynamical defects suggest additional opportunities for engineering tunable entanglement and excitations—likely relevant for measurement-based quantum computation in SPT phases with enlarged symmetry.

Future directions include:

• Mapping the continuum limit, especially at criticality, and characterizing universality classes.
• Explicit construction of particle-like excitations, their braiding, and exchange statistics.
• Connecting with integrable structures and exact solutions where possible.
• Extension to other symmetry groups and higher-dimensional SPT classifications.

Conclusion

This work provides a comprehensive constructive analysis of boundary Hamiltonians in 2D Potts SPT phases protected by $\mathbb{Z}_N$2 symmetry. The models are classified and organized by their arithmetic properties, with all phases reducible to a pyramid of primary models supplemented by dynamical defects. The explicit realization of a nontrivial 't Hooft anomaly at the boundary, together with the identification of TL algebra structures, underpins the theoretical and practical importance of these results for future studies of topological phases, quantum computation, and exactly solvable systems.

Reference:

"The $\mathbb{Z}_N$3 symmetry protected boundary modes in two-dimensional Potts paramagnets" (2604.00910)