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Two-dimensional topological paramagnets protected by Z3 symmetry: Properties of the boundary Hamiltonian
by Hrant Topchyan, Vasilii Iugov, Mkhitar Mirumyan, Tigran S. Hakobyan, Tigran A. Sedrakyan, Ara G. Sedrakyan
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Submission summary
| Authors (as registered SciPost users): | Vasilii Iugov · Ara Sedrakyan · Tigran Sedrakyan · Hrant Topchyan |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2312.15095v3 (pdf) |
| Date accepted: | Feb. 11, 2025 |
| Date submitted: | Feb. 5, 2025, 9:39 p.m. |
| Submitted by: | Sedrakyan, Tigran |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We systematically study gapless edge modes corresponding to Z3 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, given by the coset SUk(3)/SUk(2) CFT at level k=1.
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Author comments upon resubmission
We thank the Referees for their positive reports.
Below, we address Report #1 by Referee 3:
- The Hamiltonian Eq. (5) describes Nontrivial SPT states created by the unitary operators U and U^+, defined in Eq.(4). Combined with the identity operator, U and U^+ form the cohomology group Z_3. The trivial state, |0>, is the ground state of the Hamiltonian (5), which has a matrix product form. The two SPT states represent are connected with each other by |0'> = U |0>, |0''> = U^+ |0> = U |0'>.
Each SPT state has edge modes that restore the Z_3 symmetry in the presence of a boundary. The corresponding boundary Hamiltonian supporting these modes is defined in Eq.(7), where V and V' are defined by values of cohomological nontrivial U (or U2) on the boundary or trivial identity. Replacing U with its topologically different U^+, we replace V and V' with their Hermitian conjugates, which doesn't change the boundary Hamiltonian. The situation is different for the energy eigenstates. The edge states are connected with each other by the operator S acting on the boundary. They are gapless but differ from each other (for example, one can think of two types of spinons or magnons in s=1/2 systems), representing different edge states of the two topologically nontrivial SPT phases.
- Using exact diagonalization for smaller system sizes (up to N=14), we have confirmed that the first excited states at the gap point have momentum 0. Hence, it supports the claim h=\bar{h}. We have included this information in the paper. However, future investigation of a larger system size is necessary.
The revised manuscript now addresses both questions. We hope these explanations are satisfactory.
Sincerely, The authors.
Published as SciPost Phys. 18, 068 (2025)