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Anomalous boundary correspondence of topological phases
by Jian-Hao Zhang, Shang-Qiang Ning
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Shangqiang Ning |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2310.19266v1 (pdf) |
| Date submitted: | March 12, 2024, 1:28 a.m. |
| Submitted by: | Shangqiang Ning |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Topological phases protected by crystalline symmetries and internal symmetries are shown to enjoy fascinating one-to-one correspondence in classification. Here we investigate the physics content behind the abstract correspondence in three or higher-dimensional systems. We show correspondence between anomalous boundary states, which provides a new way to explore the quantum anomaly of symmetry from its crystalline equivalent counterpart. We show such correspondence directly in two scenarios, including the anomalous symmetry-enriched topological orders (SET) and critical surface states. (1) First of all, for the surface SET correspondence, we demonstrate it by considering examples involving time-reversal symmetry and mirror symmetry. We show that one 2D topological order can carry the time reversal anomaly as long as it can carry the mirror anomaly and vice versa, by directly establishing the mapping of the time reversal anomaly indicators and mirror anomaly indicators. Besides, we also consider other cases involving continuous symmetry, which leads us to introduce some new anomaly indicators for symmetry from its counterpart. (2) Furthermore, we also build up direct correspondence for (near) critical boundaries. Again taking topological phases protected by time reversal and mirror symmetry as examples, the direct correspondence of their (near) critical boundaries can be built up by coupled chain construction that was first proposed by Senthil and Fisher. The examples of critical boundary correspondence we consider in this paper can be understood in a unified framework that is related to \textit{hierarchy structure} of topological $O(n)$ nonlinear sigma model, that generalizes the Haldane's derivation of $O(3)$ sigma model from spin one-half system.
Current status:
Reports on this Submission
Strengths
2- The physical picture of the authors is new to my knowledge.
3- There are various examples that exemplify the correspondence the authors propose.
Weaknesses
2- There are a few typos or grammar errors. See suggested minor improvement.
Report
Requested changes
Below is a list of clarification suggestions the authors may consider:
1- Can the authors clarify what an “LSM anomaly” is? Presumably, this is related to the Lieb-Schultz-Mattis theorem. However, I cannot extract the definition from the manuscript or the references cited.
2- Above Eq.(6), it is said: “Now we show that for a topological order \mathcal{C}, when $\eta T = −1$, then we can also have $\eta M = −1$ for proper mirror symmetry realization. “
I was not 100 percent sure what the authors meant by “can also have.” Does this follow from an explicit construction, or does it follow from the abstract theory of classification that we can do it? From the context, I would guess the authors meant the latter. It would be nice if this could be made clearer.
3- Below Eq.(7), the authors wrote: “Now we can first identify $T_a^2 =\mu_a$ for $a\in A$. If we can further identify $T_a^2 = \mu_a$ for $ a \in B_1$, we can immediately have $\eta T = \eta M$. “
What does the word "can" mean in this sentence? Again, based on the context, I imagine the authors mean we must be able to find such an identification based on the abstract theory of classification. Is my understanding correct?
4- In the section of III.C titled “Other examples,” each example starts with a sentence (claim) about the bulk classification. Such as “For 3D bosonic topological insulators with $U(1) × Z_2^T$ and their crystalline counterparts—topological crystalline insulator with $U (1) × Z_2^M$ , their classification are both $(Z_2)^4$. ”
As no references were cited, the readers might get the first impression that this is a claim to be explained by the authors’ theory. After a more detailed reading, I believe these results (in the first sentences) are known in previous literature, and the authors use them as the basis to study boundary theory. If this is the case, the authors may consider the clarity of this situation. I will let the authors decide how to improve. I apologize if I misunderstood the purpose of the first sentences (in each example of section III C.)
Below are a few minor grammar errors (typos).
1- Page 1: can be realized many by different phases -> can be realized by many different phases.
2- Left bottom of page 4: defined to the -> defined to be the.
3- Page 7: can by done -> can be done.
4- Bottom of Page 8:“The classification for 3D topological phases protected by $SO(3) × Z_2^{T (M)}$ is classified by $(Z_2 )4$. “ The sentence structure seems not optimal.
Recommendation
Publish (meets expectations and criteria for this Journal)
Strengths
- Previously, the bridge between crystalline SPT (protected by crystalline symmetry, e.g., mirror, rotational etc.) and internal SPT (protected by global and internal symmetry) was established in the bulk by the equivalence principle. As shown in Fig.1 of this manuscript, the author steps further to consider the equivalence principle of boundary theories of the two sides. This is an innovative point.
- Importantly, the authors study the connection between the two boundary theories by considering the 3D bulk, which raises significant challenges for analytic exploration. The reason is that the 2D boundary of 3D topological phases (SPT here) usually supports diverse candidates of anomalous surface states, e.g., the so-called surface-topological-order that was previously found during the exploration of interacting-topological-insulators of both bosons and fermions. Therefore, the authors study both gapped boundaries and gapless critical surfaces, in order to reach a more complete study.
- The authors used various analytic ways to achieve the main results shown, pictorially in Fig.1, including anomaly indicators of both crystalline SPT and internal SPT, NL$\sigma$M, etc., connecting different strategies.
Weaknesses
- English writing and grammar should be improved. 2.The work, as indicated by authors in Introduction, relies on the authors' study in Ref.77 which is now a preprint and still in unpublished status. I wonder if it is possible to put these two papers together to make a systematic work for 2D and 3D. This is only a minor suggestion.
Report
Requested changes
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Above eq.2, the citation to Ref. 99 (Teo and Kane's coupled wire construction of non-Abelian quantum Hall states) is possibly a mistake. Please double check. Also above eq.3, the same thing.
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On page 3, "1. Exact anomaly". The authors stated that the manuscript avoids the discussions on the potential possibility of anomaly of emergent symmetry. I wonder if the authors can provide more statements on how to avoid this issue, say, in the following analytic exploration (both anomaly indicators and field theory). Is this an in-principle analytically solvable problem or a very hard problem?
Recommendation
Publish (meets expectations and criteria for this Journal)
We thank the referee a high evaluation on our paper . Below are our replies to the comments and questions.
Weakness: 2.The work, as indicated by authors in Introduction, relies on the authors' study in Ref.77 which is now a preprint and still in unpublished status. I wonder if it is possible to put these two papers together to make a systematic work for 2D and 3D. This is only a minor suggestion.
Reply:
We thank the referee for this excellent suggestion and we decide to follow. So we merge the two paper in the new manuscript, which now is more systematic.
Above eq.2, the citation to Ref. 99 (Teo and Kane's coupled wire construction of non-Abelian quantum Hall states) is possibly a mistake. Please double check. Also above eq.3, the same thing.
Reply:
Thanks the referee for careful reading. Indeed it was a mistake. What we want to cite is Ref.[107] in the new manuscript. We correct properly in the new manuscript.
On page 3, "1. Exact anomaly". The authors stated that the manuscript avoids the discussions on the potential possibility of anomaly of emergent symmetry. I wonder if the authors can provide more statements on how to avoid this issue, say, in the following analytic exploration (both anomaly indicators and field theory). Is this an in-principle analytically solvable problem or a very hard problem?
Reply:
We thanks the referee for asking this question. About the anomaly of emergent symmetry, there is a few example, such as the SO(5) symmetry in critical point as discussed in the paper https://arxiv.org/pdf/1703.02426 (See Fig.2 and 3 there). The SO(5) symmetry is anomalous there. But on the UV lattice model, there is no such large symmetry, but likely SO(3) together with a few discrete symmetries. So we can avoid this emergent symmetry and its anomaly by adding UV symmetry allowed terms in the surface Hamiltonian to turn away the specific critical points. In this paper, we focus only on those anomaly phenomena that is robust again the UV symmetry allowing operation.
In our opinion, the anomaly of emergent symmetry is a very hard problem, and we do not know any in-principle analytical solution at this moment. It is a very interesting question that inspire us to explore further in the future.
Author: Shangqiang Ning on 2025-12-28 [id 6188]
(in reply to Report 2 on 2024-08-04)We thank the referee for a high evaluation on our paper. Below are our replies to the comments and questions.
Reply: We thanks the referee for pointing out this confusion. By LSM anomaly, we indeed relate to the Lieb-Schultz-Mattis (LSM) theorem. In the past decades, people reinterpret the LSM theorem as an ’t Hooft anomaly matching condition, which, to the best of our knowledge, was first formulated in the Ref.[48] where the systems with the conditions of LSM theorem can be viewed as the boundary of the weak SPT. By anomaly matching condition, the boundary of the weak SPT should have the corresponding ’t Hooft anomaly. This is the basic idea for reinterpreting the LSM theorem as a ’t Hooft anomaly. In fact, along this line of thinking, many followup works have been done, among which I list a few of them here except those cited in the manuscript.
https://www.scipost.org/SciPostPhys.15.2.051/pdf https://arxiv.org/pdf/2308.00743
Reply: Thanks for the pointing out this question. In fact, by “can also have”, we mean the former meaning pointed by the referee, that is we are able to construct a symmetry enrichment pattern (including the symmetry permutation on anyon types and symmetry fractionalization) that result in the $\eta_\mathcal{M}=−1$. In fact, this is one of the advantages of the correspondence we proposed here, at least for this specific example, that is, we do not have to delve into the whole abstract theory of symmetry enrichment and exhausting all possible symmetry enrichment patterns, from which we find the corresponding anomaly pattern of symmetry enrichment.
Reply: Thanks for the careful reading and pointing the confusion in our statement. Let us clarify them here. The first “can” in “Now we can first …” means that we must be able to find such an identification based on the abstract theory. Indeed, such an identification is explicitly listed as follows (we rephrase it based on the main text): The permutation of anyon $\rho_{\mathcal{T}}$ and $\rho_{\mathcal{M}}$ are related by $\rho_{\mathcal{M}} = \rho_{\mathcal{K}} \rho_{\mathcal{T}}$ where $\rho_{\mathcal{K}}$ is the complex conjugate by mapping a to its inverse $\bar{a}$ while keeping all topological quantities invariant. (We note that $\rho_{\mathcal{K}}$ exists for any topological order described by a unitary modular tensor category.) For the symmetry fractionalization, we identify them: $\mathcal{T}_a^2=\mu_a$ for all $a$ in $C$. Specifically, for anyon $a$ in $A$, its inverse is itself, so $\rho_{\mathcal{K}}(a)=a$, behaving as no permutation. As we review our identification, we notice one previous paper which also notice such as identification: https://arxiv.org/pdf/1706.09464. We modify this identification in our manuscript and cited this reference properly.
Reply: Thanks for the careful reading and kindly reminding the citation. Yes, this classification is not our results in this paper. This complete classification actually is first obtained in Ref.[19] and using the cobordism theory in this reference https://arxiv.org/pdf/1404.6659. We note that the paper Ref[7] using group cohomology only gives $Z_2^3$, which lacks the so-called beyond group cohomology state discovered in Ref.[19]. We add the citation there properly.