Hybrid symmetry class topological insulators

We thank the referee for the report. The “Strengths” of the manuscript, “clearly written” and “Could be of interest”, as identified by the referee, are encouraging to the authors.

In response to Weaknesses 1 “Claims made … presented”, we point out that a general principle of constructing HSCTI is proposed in the Abstract and Introduction, which we exemplify with several examples in 3D, including crystalline symmetric systems and superconductors, and in 4D (Appendix E). As such, we could not see such a pedagogical approach as a “Weakness” of a manuscript. For comparison, consider the original paper on higher-order topology (HOT) Science 357, 61 (2017), which outlined a general criterion of realizing HOT by considering only two models (one in 2D and one in 3D). Later, the same principle was invoked to identify a plethora of HOT phases, including superconductors. In this work, we have presented ample examples to establish the general principle of constructing HSCTI in any dimension.

In response to Weakness 2 “Analyses do not … procedure”, we point out that we considered AZSCs for insulators and superconductors, and included phases protected by crystalline symmetries. So, which “state-of-the-art” classification procedure was missed? Different researchers follow complementary approaches for topological classification. It is impossible to commit to all such approaches in one work. We followed the canonical approach based on symmetries, namely the non-spatial time-reversal, particle-hole, and chiral or sublattice symmetries, and subsequently included the crystalline symmetries.

In response to Weakness 3 “Outline of … AZ classes or crystalline invariant”, we were wondering besides AZ and crystalline symmetry classes what other symmetry classes are there, which we might have missed? We are not aware of any other symmetry classification.

In the first paragraph of the “Report” the referee compactly summarized our key results that gave us confidence regarding the clarity of the presentation. Below we respond to the referee’s specific comments and concerns and make necessary changes to the manuscript.

Firstly, we never claimed to invent any “new class”. It is stated in the Abstract while defining the HSCTI “A fusion of two different AZSC topological insulators (TIs) such that they occupy orthogonal Cartesian hyperplanes and their universal massive Dirac Hamiltonian mutually anticommute. The boundaries of HSCTIs can also harbor TIs, typically affiliated with an AZSC different from the parent ones.”, which confirms that the Hamiltonian for HSCTI also belongs to one of the AZSCs. But the AZSC of HSCTI does not support non-trivial AZ topology as supported by examples. Namely, we considered a fusion between class AII 2D QSHI and class BDI 1D SSHI. But the resulting 3D HSCTI belongs to class A that does not have any non-trivial AZ topological invariant. We could not agree more with the referee that any Hamiltonian in any dimension belongs to one of the AZSCs, even when we invoke crystalline symmetries. But that does not imply that every AZSC supports non-trivial topology in every dimension. This is where our construction of HSCTI is novel that allows non-trivial topology beyond the predictions from AZSC. We believe that the referee possibly missed the definition of the HSCTI, which we clarify in the Abstract and Introduction.

HSCTI involves two TIs each coming with its own Wilson-Dirac mass. For such a case our proposed topological invariant from Eq. (5) is operative. In this work, we considered the universal models for QSHI and SSHI which successfully captured all their topological properties, and constructed HSCTI from their hybridization. So, we do not see which aspect of HSCTI is a “model artifact”. Topology can be established from multiple approaches, which include K-theory and homotopy, for example, as the referee pointed out. But we do not have to and cannot pursue all such approaches in one work. If our proposed definition of the topological invariant encounters any shortcoming then the referee should point it out kindly. Any lattice-based Hamiltonian bears some reductant symmetries, such as mirror and rotational symmetries, as is the case for the lattice-regularized BHZ model for QSHI, which does not invalidate the robustness of the topological phases. Furthermore, robustness of our proposed HSCTI has been established by breaking the cubic symmetry and considering the effects of disorder on the quantized surface Hall conductivities. Finally, as a matter of fact, topological invariants of all the AZSCs were computed from the Wilson mass, which later were by K-theory and homotopy, which can also be pursued for HSCTI in the future. However, it is impossible to pursue all approaches in one paper by one research group. As an example, let us count the number of years, research articles, and independent research groups it took to fully understand the ten-fold AZ classification.

We could not appreciate the criticism on the bulk-boundary correspondence, as we computed a bulk-topological invariant and showed that when it is non-trivial there are surface edge modes. Also, we established their robustness against disorder. And we once again stress that a general protocol has been put forward to construct HSCTI in the Abstract and Introduction, which then we exemplified it with a few concrete models. This is exactly how a new concept gets introduced. In this context, we point out that following our work, the proposal for HSCTI has been generalized in arXiv: 2410.18015, going by the name “Boundary topological insulators and superconductors”.

Referee’s “Requested change” for the “title” clearly resulted from the misinterpretation of the word “hybrid symmetry class”, which we have clarified in the rebuttal and revised manuscript. In response to referee’s next “Requested change” for the “general claim” we stress that in this work we have substantiated this claim with multiple examples in 3D and one example in 4D. Also, our proposed construction of HSCTI has now been further explored in arXiv: 2410.18015. Finally, in response to Referee’s last “Requested change” for the “claim of bulk-boundary correspondence”, notice that we proposed a general topological invariant for TIs involving two Wilson-Dirac masses which applies to any Hamiltonian for HSCTI and showed that when it is non-trivial the system supports robust boundary modes, thus meeting the canonical definition of the bulk-boundary correspondence. We will happily revise this discussion if the referee points out an explicit shortcoming of this correspondence. In brief, the topology of a gapped state and its bulk-boundary correspondence cannot be destroyed by infinitesimal perturbations, unless its symmetry is compromised, as shown in by considering the effects of disorder and breaking cubic symmetry.

Changes in the revised manuscript (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.

We thank the referee for the report. The referee finds our detailed study and the disorder effects as the “Strengths”, which gives authors confidence regarding the depth of our study. One of the “Weaknesses”, namely “The manuscript … not clearly written” contradicts the comment from two other referees, who praised the clarity of the presentation. In response to referee’s comment in this context “The topological invariant is discussed quite succinctly”, we point out that we discussed the topological invariant in necessary details (as per required) after introducing the model and its phase diagram, as it should be. If the referee has any specific question about the topological invariant, we request him/her to kindly specify it. We will happily address it and expand the discussion. The M phase is defined in the second last paragraph right before Sec. 3. And the XY phase is mentioned at the beginning of Sec. 3 after introducing the corresponding d-vector. Namely, in the XY phase the band-inversion occurs at the X and Y points of the BZ simultaneously that are connected by 4-fold rotations, which we now state clearly at the beginning of Sec. 3. Below we respond to the “Other concerns” by the referee and make appropriate changes to the manuscript.

In the first paragraph of the report the referee compactly summarized our key results, giving us confidence regarding the clarity of the presentation. Despite considering the referee’s suggestion to compare our work with the existing ones as a constructive criticism (which we do nonetheless), we want to point out that this is not a review article and to the best of our knowledge there is not a single model Hamiltonian for 3D systems that on the surface produces “integer” quantum Hall effect. Thus, the novelty of this work should stand on its own ground.

1. In PRB 100, 115126 (2019) authors stack layers of 2D Chern insulator in the z direction, see Eq. (10), which are imbedded within slabs of trivial insulators, see Fig. 10. However, they do not take Chern insulators in different plane and make the corresponding Hamiltonian to anticommute, which is the definition of “hybridization”, a term that we introduced. We cite this paper as Ref. 36 and add a comment on it to contrast this work with ours.

2. While mentioning axion insulators, we should have said that it can also feature a surface “half-quantized” quantum Hall effect, which we now do. Notice that ferromagnetism lifts the Kramer’s degeneracy of bands which is not the case here and the quantized Hall conductivity does not stem from any surface engineering, it is a hallmark of the bulk HSCTI. Also, “quantized” and “half-quantized” Hall effects are two distinct phenomena that shall not be compared. Notice that pi (zero) axion angles can only give half-quantized (trivial) Hall conductivity, besides non-quantized values, but not an integer-quantized Hall conductivity. See Fig. 8 of PRB 98, 245117 (2018), showing that surface Hall conductivity of axion insulators can be either half-quantized or non-quantized. Finally, we never claimed that only HSCTI features surface-quantized Hall conductivity. But to the best of our knowledge there exists no other model in the literature, showing such a phenomenon.

3. As pointed out in the last paragraph that axion angle cannot give integer-quantized surface Hall conductivity. Hence, such a phenomenon for HSCTI is not captured by an axion angle.
The referee correctly pointed out that HSCTI involves inversion of two Wilson-Dirac masses. Although this is the case for HOTIs, the inversion of two Wilson-Dirac mass does not necessarily imply HOT. As such, our model Hamiltonian for HSCTI does not support any quadrupolar or octupolar moment, signature topological invariants of HOTIs.

Next, we respond to the “Minor changes” suggested by the referee.

1. We are confused by the question from the referee “How exactly is the localization of the edge states computed? (red-blue color in Figs. 3 and 9)”. We numerically find the eigenvalues and eigenvectors, and from the spatial profile of the eigenvectors we identify its localization. Prior to the publication of this article, we will make all our codes and results available for open access via Zenodo, as we did for multiple papers in the past.

2. In all numerical transport calculations in disordered systems, we perform disorder averaging, and all the quantities have error bars as mentioned in the captions of Figs. 7 and 8. In the weak and moderate disorder regimes such error bars are extremely small (thus a bit hard to see). In the strong disorder regime, they are more visible. See Fig. 7(c) and Fig. 8 (a).

Changes in the revised manuscript (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.

We thank the referee for the report. The “Strengths” of the manuscript, “well-written”, “topical”, and “broadly applicable”, as identified by the referee, are encouraging. The referee concisely summarized our keys results, giving us confidence regarding the clarity of the presentation. We thank the referee for considering the manuscript to be “suitable for SciPost Physics”. Below we respond to the concerns of the referee, which also address the “Weaknesses” identified by the referee and make adequate changes to the manuscript.

1. We agree that any Hamiltonian in any dimension must belong to one of the AZSC. See a similar comment from Referee 2, our responses to it, and changes we make to the manuscript. We also agree that AZSC does not apply to TIs that support topological modes living on the boundary of a boundary, which we now clarify.

2. If we go by the geometry or co-dimension of the boundary modes of HSCTI, then it is tempting to conclude that 3D HSCTI corresponds to a second-order TI, featuring 1D edge modes with co-dimension 3-1=2. See also a comment from Referee 3, our responses to it and changes to the manuscript. But a HOTI must have a non-trivial higher-order moment, like quadrupolar or octupolar moment. But our HSCTI model does not possess any such moment. Thus, it is not HOTI. Notice that if we classify TIs solely based on the geometry of the boundary modes then 3D axion insulators are also second-order TIs, as they support surface edge modes, which is not the case.

3. Our protocol for HSCTI is general and it works for any combination of two TIs if the definition “A fusion of two different AZSC topological insulators (TIs) such that they occupy orthogonal Cartesian hyperplanes and their universal massive Dirac Hamiltonian mutually anticommute” is met, see the Abstract. Then we supported this claim with a combination of two strong TIs, topological superconductors, and crystalline insulators in 3D in great details, and in 4D in Appendix E. Please compare with the original paper on HOTI: Science 357, 61 (2017). They wrote only two models, and subsequent works generalized this idea to hundreds of different systems. More than a year after our work appeared on arXiv, our proposal has been generalized in arXiv: 2410.18015 (Ref. 59).

We respectfully disagree with the comment that the topological invariant should be Z not Z2, in support of which the referee quotes our results for crystalline HSCTI with in-plane C4 symmetry from Sec. 4. Consider crystalline QSHI (valley or XY phase) in Table 1 of Nat. Phys. 9, 98 (2013). In the XY phase, the parity eigenvalues at the X and Y points are -1, while at the Gamma and M points they are 1. Thus, their product from all TRIM points is +1. But that does imply that crystalline QSHIs possess a Z invariant. It continues to be a Z2 invariant. It happens generically for crystalline topological phases with Z2 invariant. Exactly, the same situation occurs with the 3D crystalline HSCTI. We clarify this issue in the revised manuscript.

In response to the final “Requested changes” from the referee, we note that in the above discussion we have argued that HSCTI should not be confused with HOTI. Our analysis covers multiple examples in 3D, involving TIs, superconductors, and crystalline TIs, and in 4D to establish the general applicability of the proposed protocol for constructing HSCTI, which has subsequently been extended to encompass other examples in arXiv: 2410.18015 (Ref. 59). It is typically the case that a new general concept is introduced with a few examples, which later gets extended and further generalized, as was the case for HOTIs in Science 357, 61 (2017). Thus, we did not make any claim that is beyond the scope of the general protocol of constructing HSCTI.

Next, we respond to the minor comments raised by the referee.

1. Indeed, in a TRS-odd system TRIM points do not have special meaning. But in lattice-based models there are two components; (a) Hamiltonian which may or may not have TRS, and (b) lattice or crystal which features high-symmetry points in the BZ that occur at the TRIM points. Even for a TRS breaking TI the band inversion occurs at the TRIM points, as is the case for QWZ model for QAHI. We now clarify this issue.

2. The term “hybrid” has already been defined in the Abstract, which we further clarify in the revised manuscript to eliminate any scope of confusion. Also, when a new term is introduced in a scientific article, such as “hybrid” in this case, readers should follow its definition given by the authors. See our responses to Referee 2.

3. The article “the” was used in the first sentence of Sec. 2 carefully, as the model from Eq. (1) captures all the outcomes we discussed in the third paragraph of the Introduction. We now expand this sentence to justify the use of the article “the”.

4. We understand that the referee and the readers would like to know more details about KWANT-based computations. We discussed key theoretical and computational steps in Appendix C. Rest of the details involving Pyhton-based codes will be made available for open access prior to the publication of this manuscript on Zenodo. We hope that the referee will understand that it is very challenging (if possible, at all) to translate Python-based codes in a manuscript.

Changes in the revised manuscript (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.