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arXiv:2512.18460 · Physical review. B./Physical review. B

Topological edge states in two-dimensional Z 4 Potts paramagnet protected by the Z 4 × 3 symmetry

Hrant Topchyan, Tigran Hakobyan, Mkhitar Mirumyan, Tigran Sedrakyan, A. Sedrakyan

Published Jun 16, 2026• 0 citations
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Abstract

We construct a two-dimensional bosonic symmetry-protected topological (SPT) paramagnet protected by an on-site G = Z 4 × 3 symmetry, starting from a three-component Z 4 Potts paramagnet on a triangular lattice. Within the group-cohomology framework, H 3 [ G , U ( 1 ) ] ≅ Z 4 × 7 , we focus on a colorless cocycle representative obtained by antisymmetrizing the basic Z 4 three-cocycle, and generate the corresponding SPT Hamiltonian via a cocycle-induced local unitary transformation H 3 [ G , U ( 1 ) ] ≅ Z 4 × 7 followed by symmetry averaging. For open geometry, we derive the boundary theory explicitly: one color sector decouples, while the nontrivial edge reduces to an interacting Z 4 chain with next-nearest-neighbor constraints that admits a compact dressed-Potts form. Using DMRG we show that the boundary model is gapless, with the lowest gap scaling as 1 / L and an entanglement-entropy scaling consistent with a conformal field theory of central charge c = 2.191 ( 4 ) ≃ 11 / 5 . The rational value c = 11 / 5 matches the coset S U ( 3 ) 3 / S U ( 2 ) 3 , making it a candidate for the continuum description of the Z 4 × 3 edge; we outline spectral and symmetry-resolved diagnostics needed to test this identification at the level of conformal towers beyond the central charge.

DOI
10.1103/xszt-gsss

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We construct a two-dimensional bosonic symmetry-protected topological (SPT) paramagnet protected by an on-site G = Z 4 × 3 symmetry, starting from a three-component Z 4 Potts paramagnet on a triangular lattice. Within the group-cohomology framework, H 3 [ G , U ( 1 ) ] ≅ Z 4 × 7 , we focus on a colorless cocycle representative obtained by antisymmetrizing the basic Z 4 three-cocycle, and generate the corresponding SPT Hamiltonian via a cocycle-induced local unitary transformation H 3 [ G , U ( 1 ) ] ≅ Z 4 × 7 followed by symmetry averaging. For open geometry, we derive the boundary theory explicitly: one color sector decouples, while the nontrivial edge reduces to an interacting Z 4 chain with next-nearest-neighbor constraints that admits a compact dressed-Potts form. Using DMRG we show that the boundary model is gapless, with the lowest gap scaling as 1 / L and an entanglement-entropy scaling consistent with a conformal field theory of central charge c = 2.191 ( 4 ) ≃ 11 / 5 . The rational value c = 11 / 5 matches the coset S U ( 3 ) 3 / S U ( 2 ) 3 , making it a candidate for the continuum description of the Z 4 × 3 edge; we outline spectral and symmetry-resolved diagnostics needed to test this identification at the level of conformal towers beyond the central charge.
Two-dimensional topological paramagnets protected by Z3\mathbb{Z}_3Z3​ symmetry: Properties of the boundary Hamiltonian
We systematically study gapless edge modes corresponding to \mathbb{Z}_3 ℤ 3 symmetry-protected topological (SPT) phases of two-dimensional three-state Potts paramagnets on a triangular lattice. First, we derive microscopic lattice models for the gapless edge and, using the density-matrix renormalization group (DMRG) approach, investigate the finite-size scaling of the low-lying excitation spectrum and the entanglement entropy. Based on the obtained results, we identify the universality class of the critical edge, namely the corresponding conformal field theory and the central charge. Finally, we discuss the inherent symmetries of the edge models and the emergent winding number symmetry. As a result, one-dimensional chains with this symmetry form a model that supports gapless excitations due to its tricritical symmetry. Numerically, we show that low-energy states in the continuous limit of the edge model can be described by conformal field theory (CFT) with central charge c=1 c = 1 , given by the coset SU_k(3)/SU_k(2) S U k ( 3 ) / S U k ( 2 ) CFT at level k=1.
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The ZN×3\mathbb{Z}_N^{\times 3}ZN×3​ symmetry protected boundary modes in two-dimensional Potts paramagnets
We construct and analyze a class of one-dimensional boundary Hamiltonians arising from two-dimensional symmetry-protected topological phases with ZN×3\mathbb{Z}_N^{\times 3}ZN×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 NNN. For prime NNN, the boundary theory admits a formulation in terms of mutually commuting Temperley-Lieb algebras. For the composite values of NNN, 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.
We investigate the continuum limit of the edge-state Hamiltonian of a two-dimensional symmetry-protected topological (SPT) model with (×Z₃)³ = Z³ × Z³ × Z³ symmetry. Starting from the lattice Hamiltonian, we first reformulate the model in terms of S3 generators and subsequently introduce a constrained Schwinger-like fermionic representation. Within this formulation, the relevant operators are expressed as fermionic bilinears, while non-local Jordan–Wigner string operators do not appear explicitly. At the critical point, λ = λ̄, we show that the continuum Hamiltonian supports gapless excitations with a linear dispersion relation, consistent with a conformal description of the edge theory. Using a mean-field treatment of the interaction terms, we further obtain a preliminary estimate of the finite-size correction to the ground-state energy. The corresponding effective central charge, c = √3 ≈ 1.73, is remarkably close to the numerical value previously obtained for the (×Z₃)³ SPT edge model and to the value predicted by the SU(3)2/SU(2)2 coset conformal field theory. Although this derivation remains heuristic, our results suggest that the constrained fermionic formulation provides a natural framework for describing the low-energy continuum limit of (Z₃)³ SPT edge states and may serve as a useful starting point for a more systematic derivation of the underlying conformal field theory.
Topological insulators in free fermion systems have been well characterized and classified. However, it is not clear in strongly interacting boson or fermion systems what symmetry-protected topological orders exist. In this paper, we present a model in a two-dimensional (2D) interacting spin system with nontrivial onsite Z2{Z}_{2}Z2​ symmetry-protected topological order. The order is nontrivial because we can prove that the one-dimensional (1D) system on the boundary must be gapless if the symmetry is not broken, which generalizes the gaplessness of Wess-Zumino-Witten model for Lie symmetry groups to any discrete symmetry groups. The construction of this model is related to a nontrivial 3-cocycle of the Z2{Z}_{2}Z2​ group and can be generalized to any symmetry group. It potentially leads to a complete classification of symmetry-protected topological orders in interacting boson and fermion systems of any dimension. Specifically, this exactly solvable model has a unique gapped ground state on any closed manifold and gapless excitations on the boundary if Z2{Z}_{2}Z2​ symmetry is not broken. We prove the latter by developing the tool of a matrix product unitary operator to study the nonlocal symmetry transformation on the boundary and reveal the nontrivial 3-cocycle structure of this transformation. Similar ideas are used to construct a 2D fermionic model with onsite Z2{Z}_{2}Z2​ symmetry-protected topological order.