SciPost Submission Page
Correspondence of boundary theories between internal and crystalline symmetry protected topological phases
by Jian-Hao Zhang, Shang-Qiang Ning
Submission summary
| Authors (as registered SciPost users): | Shangqiang Ning |
| Submission information | |
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| Preprint Link: | scipost_202512_00055v1 (pdf) |
| Date submitted: | Dec. 28, 2025, 1:02 p.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
{Symmetry protected} 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 a correspondence directly in two scenarios, including the anomalous symmetry-enriched topological orders (SET) and critical {boundary} 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. {In this perspective, we first consider the edge-corner correspondence between edge theory as 1+1D conformal field theory of internal fermionic SPT and the 0+1D corner modes of (higher-order) crystalline fermionic SPT. By viewing the corner modes on 1D boundary as perturbed CFT is crucial insight for the correspondence, but also help to discover the boundary theory of some intrinsically interacting fermionic SPT, whcih are challenging.} {Next, we discuss this correspondence bosonic systems, 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 {(NL$\sigma$M)}, that generalizes the Haldane's derivation of $O(3)$ sigma model from spin one-half system. {To our best of knowledge, this hierarchy structure of the $O(n)$ NL$\sigma$M is first exposed, which inspires us to explore more interesting connection between different theories in the light of the boundary correspondence. }
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
We thank the referees very positive reports. Both the referees consider our works of high originality and good or high validity. Both the referees asked a few questions or point out some issues about the presentation or understanding of some concepts in the manuscript. We replied to them properly in the revised manuscript. In particular, the referee 1 had noticed that our another related work (Ref[77] in the previous manuscript and the corresponding preprint number is arXiv:2112.14567) on this topic has not been published and he/she has suggested “if it is possible to put these two papers together to make a systematic work for 2D and 3D.” We consider this is an excellent suggestion and decide to follow. So we merge the two works in the new manuscript properly. Now we hope the revised manuscript is more systematic and is ready for publication in SciPost.
Yours Sincerely,
Jian-Hao Zhang and Shang-Qiang Ning
List of changes
- Change the title to be more specific
- In the abstract, the main change is that after the sentence “(2) Furthermore, we also build up direct correspondence for (near) critical boundaries.”, we add “In this perspective, we first consider the edge-corner …systems, again” and in the end, we add “To our best of knowledge, this hierarchy … correspondence.”
- In line 184, we add “, and also to the fermionic SPT”.
- We add the content from line 299 to line 331.
- We delete the content (below Eq.7 in the first manuscript and the Eq. 43 of the new version) from “Now we can first identify… while keeping all the topological quantities invariant” and add the new content “To make connection between…leads to $\eta_\mathcal{M}=-1$.”.
- We add a new section 3, from line 402 to line 559
- We add the Appendix A, B, C from line 963 to line 1021
- Add citations [19,114] at the end of the first sentence below Sec.4.3.1.
- Change the citation above eq.38 (i.e., eq. 2 in the previous manuscript) properly.