Hybrid symmetry class topological insulators

1. Clearly written
2. Could be of interest

1. Claims made as with respect to results presented
2. Analyses do not reflect current state-of-the-art understanding of classification procedures
3. Outline of conditions, i.e. are the authors considering AZ classes or crystalline invariants

In this paper Das and Roy claim to introduce a so-called “hybrid symmetry class” of topological insulators. They consider a Qi-Wu-Zhang kind of model on one plane and a SSH kind of model in the remaining Z direction. As one can pick anti-commuting matrices one part of the hybrid Hamiltonian acts as a mass term for the other model and vice versa. The authors also consider the edge state spectrum and behaviour under defects and disorder.

The manuscript reads easily. However, one should be concerned with some of the scientific statements.

First of all the claim of inventing a “new class” is severely out of place. The tenfoldway can be connected to K-theory and is exhaustive provided one asks the right question. That is, once one only considers particle hole symmetry, time reversal and chiral symmetry all possibilities are given by the ten fold way. Here the authors take two orthogonal planes for the two parts of the hybrid Hamiltonian and hence already assume translational symmetry [this is crucial and for example induces weak invariants in a 3D class AII TI]. On top of that they later consider parity and C_4 symmetry. From the perspective of the AZ classification one adds two terms A and B the symmetry of both classes together will determine the AZ classification and under those conditions this is exhaustive, i.e. there is no room for a new class. If one takes into account crystalline symmetries this can be enhanced but the 3D phase still will exhaustively belong to an AZ class with respect to those symmetries. As such the title and some statements are misleading and need to be updated.

Secondly, the authors claim generality but only present a very simple model. Indeed, both parts of the Hamiltonian have many redundant symmetries due to their simplicity [such as a mirror and rotational symmetries]. Such models are fine to illustrate the physics, but if one claims general results a general topological invariant should be presented. No such invariants are given other than the WIlson mass. The latter is fine to tune models but is not related to the possible topological indices without extra steps. If this is a hybrid class in what sense? There are basically only three options: Either they are stable phases and can be captured by K-theory or they can be captured by homotopy theory or they are model artifacts. In this regard I also note the improvised construction although by now several strategies can be used to generally evaluate crystalline TIs.

The same criticism holds for the claimed bulk-boundary correspondence. As before I think a correspondence can only be claimed if there is a direct invariant and a relation to edge does, e.g. as with anomaly inflow arguments for Chern insulators. Apart from a specific model calculator no correspondence in general terms can be concluded from the arguments in the paper. With regard to the stability of the modes [i.e HOTI] discussion is is clear that a symmetry analysis can resolve this as these will not be intrinsic [i.e. due to the SSH part they are more like obstructed TI edge modes].

As such this paper requires drastic changes before publication can be recommended.

1. title
2. general claims
3. claim of bulk boundary correspondence

Ask for major revision

1. The authors go in great detail to describe the models at hand.
2. It is nice to see that the effect of disorder has been considered in their transport calculations

1. The manuscript is at times not very clearly written. The topological invariant is discussed quite succinctly. In section 3, they refer to the M and XY phases, which never seem to be define them in the main text (unless I missed them). The authors promise in section 3 they will define them in section 4, but they are only to be found in Appendix D.

2. Other concerns regarding the content are described in the requested changes.

The authors propose a hybrid class of topological insulator states. The proposal is based on constructing a Hamiltonian based on two lower-dimensional topological insulators belonging to different Altltand-Zirnbauer classification class. The authors describe toy models of this phase, including crystalline and superconducting versions, perform transport computations to confirm surface-state physics and propose an invariant.

The work and calculations within it seem sound. The concept is interesting. However, to be able to give a recommendation for this paper it would be helpful to have the authors clarify certain points concerning the differences
with existing work. I detail these points and others that I would like the authors to address below.

1. I would like the authors to discuss the similarities and differences with the related concept of Embedded topological insulators, see "Embedded topological insulators" by Tuegel et al. Phys. Rev. B 100, 115126 (2019). At first I was worried that this was the same construction, but here the authors seem to mix symmetry classes, while Embedded topological insulators mix topological insulators of the same symmetry class. A discussion that acknowledges the differences with the idea in this work seems relevant.

2. After reading the manuscript, it is still quite unclear to me why this state is not like a Ferromagnetic, inversion breaking, axion insulator. The authors say that axion insulators display a non-quantized surface conductivity. However I do not think this is generically true. Their comment could refer to 3D TIs that have gapped top and bottom surface states, with half-quantized surface conductivity, like in Ref [33], in which case I believe their comment applies. However, more generally, an axion insulator that breaks T and I could have an integer surface Hall conductivity, if all surfaces are gapped but their gaps have a relative sign difference. Consider as an example a TI without inversion where a bulk Zeeman is chosen to have opposite signs on the side surfaces compared to the bottom surfaces. A construction like this appears for example in Varnava and Vanderbilt Phys. Rev. B 98, 245117 (2018), Fig. 9c) . In this example having the side colors "blue" while the top being "red" of a cubic system seems to realize what is shown in the present paper in Fig. 1aiii . How is this different?

3. I am confused by the bulk invariant, and its relation to other known invariants. First, in-line with comment 2, I think the three-dimensional bulk invariant should be related to the axion angle. Second, if I understood correctly, the invariant is counting two inversions, that of the QSH and that of the SSH. This resembles a second-order topological invariant, where two inversions are required to define the state. Both points are not mutually exclusive, since I beleive axion insulators can be thought of as T- or I-broken HOTIs.

Minor changes:

a. How exactly is the localization of the edge states computed? (red-blue color in Figs. 3 and 9). The authors should be more precise.

b. For the disorder calculations, the authors average over disorder realizations. This defines an error bar to the calculation, the standard deviation, that they could add to Fig. 8.

Ask for major revision

1. well-written
2. topical
3. broadly applicable

1. presentation / claims misleading in several points
2. detailed example but no general proof

The authors introduce the concept of "hybrid symmetry class topological insulators" and work out one particular example in detail.

The manuscript is very well-written, and the analysis of the example is both clear and thorough.
Moreover, the idea of combining two types of topological insulators to create a third is certainly useful and seems versatile and broadly applicable in follow-up work.
The content is therefore suitable for SciPost Physics.

However, I believe the way this idea is presented in the current manuscript is misleading in several central aspects.
I would like to suggest that the authors phrase these differently.

In particular:
1) the authors claim several times that their results go beyond the Altland-Zirnbauer classification.
However, this is simply not true, and moreover, not possible within the current setup. The Hamiltonian of Eq. (3) is a 3D Hamiltonian for non-interacting spin-1/2 particles in a two-orbital lattice. As written by the authors themselves below Eq. (4), this Hamiltonian falls in class A of the Altland-Zirnbauer classification, which in 3D is always trivial. This agrees with the observation of the authors that introducing a single (infinitely large) boundary into the periodic system along any direction results in an absence of surface states. The Altland-Zirnbauer classification does not say anything about the boundaries of boudaries (see also point 2 below), so as far as I can see the model introduced by the authors neatly fits into the Altland-Zirnbauer paradigm.

2) The authors also claim that the model of Eq. (3) is not a higher-order toplogical insulator (HOTI). However, introducing a boundary of the boudary (that is, cutting the periodic system in two orthogonal directions) may result in the emergence of edge states along the hinges. This clearly shows, as also argued by the authors, that the 3D system is a trivial insulator, while its 2D surface Hamiltonian (in one direction) is a topological insulator. A trivial insulator whose edges are topological insulators is the textbook definition of a second order topological insulator. I therefore do not understand why the authors insist their model is not a HOTI.

3) The authors present their results in very general terms, suggesting that all results are generic and that their construction will work for any combination of topological systems. However, they analyse only a single model in detail and discuss broader applications only in terms of extensions of that one model. There is no proof that any of the presented results are applicable more generally. In fact, some results certainly are not. For example, the topological invariant of eq. (5) should in general be Z-valued, rather than Z2-valued (being an invariant for a QAHI), and already fails to apply to the system with C4 symmetry in section 4 (as mentioned by the authors).

I would suggest that the authors are open about these aspects: their work provides a methodology for constructing HOTIs from lower-dimensional topological systems, and they analyse one particular example in great detail. This is a worthwhile result, without the need to claim anything more.

I also noticed some minor details:

- In the introduction, band inversions at TRIM are mentioned in a sentence referring to all AZ classes. Since TRIM do not have any special meaning in TRS-broken classes, this statement should be revised.
- The terminology "hybrid" can be confusing: the authors refer to hybridization between two terms in the Hamiltonian, rather than hybridization between spatially separated systems with an interface. It would be good to make this explicit early on.
- In the first sentence of section 2, the author mention "the" model, where they probably mean "a" model.
- It would be good to include details of the KWANT algorithm in appendix C.

- Rephrase the presentation of the results to avoid misleading claims.
- Consider minor points mentioned in the report.

Ask for major revision