SciPost Submission Page
Hybrid symmetry class topological insulators
by Sanjib Kumar Das, Bitan Roy
Submission summary
| Authors (as registered SciPost users): | Bitan Roy |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202403_00040v2 (pdf) |
| Date accepted: | June 9, 2025 |
| Date submitted: | April 23, 2025, 7:08 p.m. |
| Submitted by: | Roy, Bitan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Traditional topological materials belong to different Altland-Zirnbauer symmetry classes (AZSCs) depending on their non-spatial symmetries. Here we introduce the notion of hybrid symmetry class topological insulators (HSCTIs): A fusion of two different AZSC topological insulators (TIs) such that they occupy orthogonal Cartesian hyperplanes and their universal massive Dirac Hamiltonian mutually anticommute, {\color{blue}a mathematical procedure we name hybridization}. The boundaries of HSCTIs can also harbor TIs, typically affiliated with an AZSC that is different from the ones for the parent two TIs. As such, a fusion {\color{blue}or hybridization} between planar {\color{blue}class AII} quantum spin Hall and vertical {\color{blue}class BDI} Su-Schrieffer-Heeger insulators gives birth to a three-dimensional {\color{blue}class A} HSCTI, accommodating quantum anomalous Hall insulators {\color{blue}(class A)} of opposite Chern numbers and quantized Hall conductivity of opposite signs on the top and bottom surfaces. Such a response is shown to be stable against weak disorder. We extend this construction to encompass crystalline HSCTI and topological superconductors (featuring half-quantized thermal Hall conductivity of opposite sings on the top and bottom surfaces), and beyond three spatial dimensions. Non-trivial responses of three-dimensional HSCTIs to crystal defects (namely edge dislocations) in terms of mid-gap bound states at zero energy around its core only on the top and bottom surfaces are presented. Possible (meta)material platforms to harness and engineer HSCTIs are discussed.
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
List of changes
Changes in the revised manuscript in response to Referee 2 (shown in blue):
1. In the Abstract and Introduction, we now properly define the mathematical definition of `hybridization’. In addition, we also explicitly mention the AZSCs of two parent TIs and the resulting HSCTI therein, so that there is no confusion with the fact that HSCTI also belongs to one of the AZSCs in Sec. 2 as well, which, however, does not feature known AZ topology, thereby supporting the novelty of the construction of HSCTI and its topology.
2. At the end of the last paragraph of `Discussions and outlooks’ section, we point out that recently our proposed protocol of constructing HSCTI has been generalized in arXiv: 2410.18015 (a new Ref.~60). Along with this preprint, multiple examples we have presented in this work should anchor the general applicability of our proposed protocol.
Changes in the revised manuscript in response to Referee 3 (shown in blue):
1. At the beginning of Sec. 4, we show the d-vector for crystalline TIs as a new Eq. (6) [previous in-text expression]. Immediately after Eq. (6), we define the XY phase.
2. At the end of the third paragraph of Introduction, we contrast our HSCTI with `Embedded topological insulators’ and cite PRB 100, 115126 (2019) as Ref. 37.
3. We now cite PRB 98, 245117 (2018) as Ref. 36, and toward the end of the third paragraph of Introduction we add a discussion on the axion angle induced surface Hall conductivity to argue that it can only give rise to half-quantized or non-quantized surface Hall conductivity, but not to integer-quantized Hall conductivity on the surfaces.
4. At the end of the paragraph after Eq. (5), we state that HSCTI does not possess quadrupole or octupole moments, hallmarks of HOT, and cite Refs. 47-49 where they were first computed.
5. In the captions of Figs. 3 and 9, we specify how to compute the localization of each mode.
Changes in the revised manuscript in response to Referee 1 (shown in blue):
1. Right before Eq. (5) and at the end of the second paragraph of Sec. 4, we mention that as HSCTI supports topological modes on the boundaries of a boundary, AZ topology does not apply there, even though its Hamiltonian belongs to one of the ten AZSCs.
2. At the end of the paragraph, following Eq. (5), we argue that HSCTI and axion insulators should not be considered as HOTIs just because they support edge modes on the surfaces. See also `Changes in the revised manuscript’ No. 4 in response to Referee 3.
3. End of first paragraph of Sec. 4: We clarify the topological invariant for crystalline symmetry protected HSCTI by comparing it with a similar phenomenon in crystalline QSHI.
4. At the end of the second paragraph of the Introduction, we state that in TRS breaking systems TRIM points do not have any special importance. But we also say that in lattice-regularized models of such systems, the band inversion still occurs around the TRIM points as they occur at the high symmetry points of the BZ.
5. We expand the first sentence of Sec. 2 to justify the use of the article `the’ therein.
Unsolicited changes in the revised manuscript (NOT shown in blue):
1. We expand the Abstract to include more important details of our study.
2. We display the roadmap of the paper as a separate subsection, Sec. 1.2 and expand it by including details of the materials covered in various appendices.
3. In addition, we made multiple cosmetic changes to improve overall lucidity of the presentation without altering any major or minor conclusions.
Current status:
Editorial decision:
For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)
Reports on this Submission
Report
I still have thoughts about the axion angle point, becauset I believe it is possible to write an axion field theory that leads to an integer Hall effect at the boundary . What is the correct field theory of this hybrid phase is, in my opinion, an open question, which I do not think should prevent publication since this is not the goal of the paper. I am thus happy with the additional comments that point out the differences between current versions of axion insulators.
To clarify one of my previous comment, localization length can be calculated in equivalent, but not numerically equal, ways (e.g. IPR or by fitting the real-space wave-function profile, as they seem to do). I am happy that they now clarify how this is done precisely and that they will share their codes.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report #1 by Jasper van Wezel (Referee 1) on 2025-5-9 (Invited Report)
Report
The additional clarification of the author's definition of "hybridization" and the added discussion in several places of AZ classes adequately resolve the original ambiguity around how HSCTI fit into the AZ classification.
The added definitions and discussion of HOTI and axion angle now also make sufficiently clear what the authors consider to be essential differences between them and HSCTI.
I would personally still prefer a slightly more careful phrasing of claims about the generality of the proposed HSCTI construction and their properties, but I appreciate the replies of the authors and don't think this should affect publication.
I believe the authors adequately addressed all issues raised, and I recommend publication.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)