Two-dimensional topological paramagnets protected by $Z_3$ symmetry: Properties of the boundary Hamiltonian

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The manuscript entitled "Two-dimensional topological paramagnets protected by
Z3 symmetry: Properties of the boundary Hamiltonian" is a submission to
SciPost Physics Lecture Notes. It is difficult to read and difficult to view
as teaching material: In the manuscript we read about coset SU_k(3)/SU_k(2),
xZ_3 and H^{d+1}(S,U(1)) cohomology groups which are not explained, but the
symbol [x] as the integer part of the number x is so.

The manuscript consists of three parts:

I) Section 1 Introduction
This presents the "theory" of symmetry-protection, topological (SPT) phases
and cohomology groups.

II) Sections 2-4
Here a rather concrete model is presented.

Section 2
The three-state paramagnetic Potts model defined on a triangular lattice is
introduced, its bulk Hamiltonian (5) and the boundary Hamiltonian (6). Then by
use of "certain transformations" the alternative form (16) is derived.

Section 3
Here symmetries of the boundary Hamiltonian are investigated.

Section 4
Yet another, alternative form of the boundary Hamiltonian is derived.

At this point the question comes up if the symmetries discussed here and the
general theory of the introduction help in the understanding of the physics of
the boundary Hamiltonian and possibly also of that of the bulk Hamiltonian? Or
is it the opposite, and the Hamiltonians (6), (16), and (36) serve as an
illustration of the "general principles" of SPT in the introduction.

If the symmetries help in the understanding of the physics of the boundary
Hamiltonian, I do not understand why a numerical analysis as in the following
section 5 is necessary.

III) Section 5
Here the conformal properties of the edge model are investigated by use of
DMRG. The authors use the CFT relations of finite size terms in the energy
spectrum to the conformal weights. Also the entanglement entropy is computed
from which the authors identify the conformal weight as c=1. Question: what do
you find from the ground state energy of the (closed and/or periodic) chain?
Does it agree with c=1? And why is the theory so different from that of the
three-state Potts CFT, also known as the parafermion CFT with central charge c
= 4 / 5?

These are just the first questions that come to my mind after reading the
manuscript. I hope the authors can clearly structure and extend the manuscript
and guide the reader through the material.

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This manuscript systematically investigated the gapless edge modes for the topologically nontrivial $\mathbb{Z}_3$ SPT phases of 2d three-state Potts paramagnets placed on a triangular lattice. Starting from the topologically trivial phase, the authors constructed the topologically non-trivial phase corresponding to a non-trivial element of $H^3(\mathbb{Z}_3,U(1))=\mathbb{Z}_3$ (which classifies the bosonic $\mathbb{Z}_3$ SPT phases in 2d), together with the boundary Hamiltonian residing on the edge of the triangular lattice. Further analyzing the symmetries of edge Hamiltonian, the authors discovered an additional winding symmetry and computed the 't Hooft anomaly for the $\mathbb{Z}_3$ symmetry of the boundary theory. Additionally, the authors invoke numerical methods such as DMRG and exact diagonalization to study the finite-size scaling of low-energy excited states, the ground state entanglement entropy, and matrix elements of the Kac-Moody current commutators and relations acting on low-energy excited states. Based on these numerical evidence, the authors conjectured that the low energy effective theory for the edge model is given by the coset $SU(3)_1/SU(2)_1$ CFT.

This manuscript presented a neat idea regarding an explicit construction of topologically non-trivial $\mathbb{Z}_3$ SPT phase and the corresponding edge Hamiltonian. Moreover, using extensive numerical studies, the authors investigated properties of the edge Hamiltonian, making a reasonable conjecture on the low energy continuum limit of the edge model. Despite the great strength of this manuscript, there are several places that are worthy of improvements. Therefore, the authors are invited to make some modifications regarding the following suggestions/comments.

1. Several notations in Section 2 deserve further comments. For example, below the definition of the unitary in Eq. (4), "color" was attributed to the nodes of triangles, do the three colors here correspond to the three sublattices? The authors are invited to comment on this. It would also be nice to supplement with a figure demonstrating the geometry. Additionally, below Eq. (5), it would be clear to specify/define $p_p$.

2. In this manuscipt, the authors focused on one of the two topologically non-trivial $\mathbb{Z}_3$ SPT phases. The authors are invited to make comments on the changes required to obtain the other non-trivial $\mathbb{Z}_3$ SPT phase. In particular, starting from a unitary slightly different from Eq. (4), the resulting edge Hamiltonian will have a different anomaly captured by the corresponding 3-cocycle.

3. In the abstract, the authors described that the two topologically non-trivial phases and the trivial phase define a general one-dimensional chain supporting a tricriticality. The authors are invited to elaborate more on this in the main text.

4. Regarding the finite-size study of the excitation gap, there seems to be a mismatch between Eq. (45) and Figure 1. From Eq. (45), setting the gap to be 1 yields the value of N to be around $4\pi$, which is much smaller than the value read from Figure 1. Also it seems that Eq. (45) can be derived from 1d chain with the periodic boundary condition instead of open boundary conditions used in DMRG. Finally, is $x_N$ the conformal dimension for the CFT operator corresponding to the first excited state? (The first Virasoro descendant states of the identity operator in the CFT already have a smaller scaling dimension.)

5. It would be nice if the authors could give a bit more supporting arguments regarding the conjectured low energy continuum limit of the edge Hamiltonian.

6. There are a few typos scattered around in the manuscript. For example, "boundary" above Eq. (36), "the continuum limit" below Eq. (47), "power law" in the caption for Figure 4. The authors are invited to double-check and correct the typos.

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Thorough investigation the boundary physics of a $\mathbb{Z}_3$ protected SPT using advanced analytical and numerical methods.

Dense presentation, lack of figures to explain the construction, close similarity to Ref [61]

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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