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Two-dimensional topological paramagnets protected by $Z_3$ symmetry: Properties of the boundary Hamiltonian
by Hrant Topchyan, Vasilii Iugov, Mkhitar Mirumyan, Tigran S. Hakobyan, Tigran A. Sedrakyan, Ara G. Sedrakyan
Submission summary
| Authors (as registered SciPost users): | Vasilii Iugov · Tigran Sedrakyan |
| Submission information | |
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| Preprint Link: | scipost_202402_00014v1 (pdf) |
| Date submitted: | 2024-02-06 22:43 |
| Submitted by: | Sedrakyan, Tigran |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We systematically study gapless edge modes corresponding to $\mathbb{Z}_3$ 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 symmetry distinguishing between two possible $\mathbb{Z}_3$ SPT phases. As a result, the two topologically non-trivial and one trivial phases define a general one-dimensional chain supporting a tricriticality, which we argue supports a gapless SPT order in one dimension. 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 $SU_k(3)/SU_k(2)$ CFT at level k=1.
Current status:
Reports on this Submission
Strengths
Thorough investigation the boundary physics of a $\mathbb{Z}_3$ protected SPT using advanced analytical and numerical methods.
Weaknesses
Dense presentation, lack of figures to explain the construction, close similarity to Ref [61]
Report
The authors investigate a boundary physics of a $\mathbb{Z}_3$ symmetry-protected topological phases of a two-dimensional Potts paramagnet on a triangular lattice. Following ideas of Levin and Gu and authors' recent work [61] they construct an edge model, investigate its microscopic and emergent symmetries and propose its effective low-energy theory. All this is done with the help of advanced analytical techniques used together with numerical DMRG approach. The paper is very similar in spirit to Ref [61], where a closely related (but different) model was investigated using the same techniques. Before providing final recommendation, I would ask the authors to address the following comments and questions:
--- Given that the model studied in Ref [61] is closely related to the model studied in the present paper, it would be appreciated if the authors provide more detailed comparison with the results of Ref [61] and argue what makes the present study qualitatively different from it to justify a separate publication.
--- To clarify presentation of the main analytical arguments, I suggest to add figures. For example, a figure related to the unitary (4) will help to clarify its construction. Also a figure illustrating construction of the edge model would be useful.
--- Given that there are three distinct SPT bulk phases in the studied problem, there should be different edge theories separating various phases. I am confused which edge theory is actually analyzed in this paper. Please clarify what is meant by the sentence in the abstract "As a result, the two topologically non-trivial and one trivial phases define a general one-dimensional chain supporting a tricriticality, which we argue supports a gapless SPT order in one dimension."
--- How is the microscopic non-abelian permutation group $\mathbb{S}_3$ embedded in the edge low-energy effective field theory?
--- Given that the $\mathbb{Z}$-valued winding symmetry only emerges in the edge model (which appears on the boundary of two different SPT phases), it is unclear to me what is meant by "it distinguishes the non-trivial phase from the trivial one" written after Eq. (25).
Minor comments: (i) Clarify what is meant by "the calculation of their investment directly" on page 5; (ii) typo Ref. Fig. 1 on page 11.
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