Topological states between inversion symmetric atomic insulators

The authors claim that a group of weak topological invariants, which depend only by the symmetries of the atomic lattice, induces a bulk-boundary correspondence.

In particular, it is claimed that (i) these weak topological invariants predict the presence or absence of states localised at the interface between two inversion-symmetric band insulators with trivial values for their strong invariants, and (ii) the interface modes are protected by the combination of band topology and symmetry of the interface.

These statements are strong and counterintuitive, and I don’t find the supporting arguments in the manuscript convincing. The essential counterargument opposing these statements that it seems likely that it is possible to introduce a perturbation which obeys the symmetries but moves the interface states in energy to the conduction or valence band on either side of the interface. Thus, the interface states hybridise with the bulk states and are no longer localised at the interface. (Notice that the energy gap can be very small on one side of the interface, and therefore intuitively it seems that it would be very easy to hybridize the interface states with the bulk bands on that side of the interface.) This would mean that the interface states are not protected by the combination of the band topology and symmetry of the interface, and therefore also the weak topological invariants would not always predict the presence or absence of localised states at the interface.

It is probably possible to formulate a weaker statement, which is related to statement (i). Namely, I expect that quite generically these weak topological invariants are related to the presence or absence of localised states at the interface. The appearance of such interface states can be understood by first considering a smooth interface with slowly varying parameters connecting the two Hamiltonians, and then realising that one needs quite large perturbation to remove the resulting localised interface states appearing inside the bulk gap. Thus, the interface states will often survive in specific models even in the case of a sharp interface. Nevertheless, I want to emphasise that these localised interface states are not protected just by symmetry and topology; Their appearance requires additional assumptions about the model Hamiltonian.

I am willing to reconsider the paper for publication if the authors have additional arguments to support their statements that the interface states are indeed protected by the combination of band topology and symmetry of the interface. For this purpose I suggest that the authors construct an explicit example (as suggested also both by Daniel Varjas and the first referee), and study the robustness of the interface states with respect to introducing all possible symmetry-preserving perturbations which do not change the band topology.

1) this is a very well-written paper
2) the splitting into main text/appendices is appropriate, and the latter provide many useful details on the authors' results
3) the results are general and apply in a model-independent way
4) I believe this work will lead to multiple follow-up theoretical and experimental investigations

1) it would really help the readers to see one simple example Hamiltonian showing the behavior discussed in the paper
2) I'm confused about the use of the term "topological protection." It might be helpful to better clarify this term.

The authors consider interface states between 2D atomic insulators in class A, which the authors define as systems with vanishing Chern number. Focusing on space groups p2, p3, and pmm, the authors show a simple and generic criterion predicting localized interface states. This is based on the logarithmic derivative of the wavefunction, which is constrained by bulk topology, namely by the symmetry labels of the bands.

I really enjoyed reading this paper, it is very well written. Results are presented in a step-by-step pedagogical way, making the authors' discussion easy to follow also for non-specialists. I think this work is timely and of interest to the community working on topological systems, and it will motivate experimental studies on these kinds of interface states, possibly in various meta-material platforms.

I only have two concerns with regard to this work, and I would like to ask the authors to please address them before I can recommend publication. I recognize that I am biased in this regard, since I have already read the question asked by Daniel Varjas on the SciPost submission page, and I fully agree with him.

The two points are:

1) Please add an example. I understand that the work is general, and that the results apply in a model-independent way: this is one of the strengths of the paper. However, it would really help the readers to have something concrete to point to while going through these general results. With even a simple example in an appendix somewhere, I believe the quality of this paper would be greatly improved.

2) Several times throughout the paper, but especially at the beginning of Section 4, the authors refer to interface states which should be present at the Fermi level. At the end of Section 5, the authors say that these interface states may sensitively depend on crystal terminations. This makes me confused with respect to the way in which the authors define "topological protection."
My confusion is as follows:
As far as I can understand (again, here an example would go a long way), the symmetry labels force interface-bound states to exist, but they don't force these states to continuously interpolate in energy between the valence and the conduction band. My naive guess (which could be wrong) is that these states exist at some particular energy, $E_p$, which is in general different from 0 since there is no chiral or particle-hole symmetry in class A. The interface between two inversion-symmetric bulks is in general not inversion-symmetric, so it might be possible to change the microscopic details of the interface without worrying about inversion breaking. In this case, isn't it possible for me to freely move these interface states in energy, simply by changing the microscopic details of the interface itself? For instance, could I add a chemical potential just to the interface, and push the interface states up in energy until they overlap with the bulk bands?
Note that this would be very different from the "topological protection" of strong TIs (e.g. quantum spin-Hall effect) and what people used to call weak TIs (e.g. 3D stack of 2D QSHE). In those systems, the boundaries are invariant under the symmetries (time reversal, translation) and the boundary states exist at the Fermi level no matter how one changes the microscopic details of the interface, provided its symmetries are left intact.
Are the states discussed in this paper and their amount of topological protection different from Shockley states? Does this paper discuss the conditions for which Shockley states exist in p2, p3, and pmm?
The authors should clarify their definition of topological protection when it comes to the constraints on the energy of interface states. Also they should clarify the connection/difference between the interface states they consider and conventional Shockley states. For instance, with a toy-model example it could be possible to show the states and see if/how they move in energy away from the Fermi level when a chemical potential is added only to the interface region.

See report.