Topology in time-reversal symmetric crystals

Dear editor,

thank you very much for your email of 19 September, advising us that, based both on the published referee reports and on private communications unseen by us, you recommend resubmission of a thoroughly revised manuscript.

Having carefully studied the reports, we feel that we can adequately address all of the concerns of the referees. We appreciate the main concern of both referees, that some of the explanations in our work lack a fully rigorous mathematical backing. In fact, the central aim of the present paper is to present a simple and intuitive picture for a topological classification of time-reversal invariant crystals that is accessible by anyone in the field, including experimental as well theoretical physicists. It is for this reason that we avoided overly formal mathematical expositions in the original submission.

Although we still feel that this is the correct choice for the present work, we appreciate the advice of the referees, and urged forward by their questions, two of us (JK and JdB) set out a collaboration with pure mathematicians to study in detail the K-theoretical calculations underlying the present work in at least some of the more accessible space groups. This has now resulted in a purely mathematical paper available on the ArXiv (https://arxiv.org/abs/1811.02592), which confirms everything we write in the present submission. Because of the highly specialised and formal nature of the calculations in this follow-up work, we chose to publish it as a separate mathematics paper rather than as an appendix to the current physics paper. Nevertheless, the results of this new work strengthen our confidence in the results of the present submission, and we cite it in the revised manuscript as well as in the response to the referees below.

We are also confident that the responses below adequately address the questions and concerns raised by the referees, and that the changes to the manuscript prompted by the referee reports add to the clarity of its presentation. We are grateful for the encouraging remarks and helpful suggestions of both referees, and respectfully ask you to reconsider our submission for publication in SciPost Physics.


Yours sincerely,

Jorrit Kruthoff, Jan de Boer, and Jasper van Wezel.




Below, we respond to the remarks of the reviewers on a point-by-point basis. The relevant text from the original review reports appears between quotation marks.


Response to the First Referee

The referee starts out by stating:
"The authors overstate the novelty of their own work while failing to give proper credit to earlier papers. The authors make the bold claim in the introduction: "The present work ... thus provides for the first time a methodical algorithm for counting topological phases in time-reversal symmetric crystals." Yet, Ref 12 earlier enumerated topological phases in time-reversal symmetric crystals. Furthermore, Bradlyn et al (Nature volume 547, pages 298–305 (2017)), which is not cited, also presents a classification for topological phases in time-reversal symmetric crystals. There are differences between each of these classifications. It is remiss to not mention the earlier papers and compare/contrast the classification schemes. This would help justify the novelty of the current paper."

We thank the referee for pointing out the lack of comparison with existing literature. We agree with the referee that this is an unjust omission on our part, and include in the revised manuscript a short discussion comparing earlier approaches to ours. We now also cite the work by Bradlyn et al. mentioned by the referee.

To be specific, our work goes beyond the earlier results of Bradlyn et al. and also that of Po et al., by including a classification and intuitive discussion of topological invariants that are not related to representations of the crystal symmetry. Notice that this not a trivial addition, since although these other invariants do not arise from the crystal symmetry, both their allowed number and type may be constrained by the crystal symmetries.

A full classification of time-reversal invariant crystals must necessarily include both kinds of invariants, as presented in the current manuscript. This assertion also follows naturally from the mathematical computations in K-theory presented in https://arxiv.org/abs/1811.02592.


The referee continues:
"Similarly, the claim, "A systematic classification of all possible topological phases in the presence of a given crystal symmetry and dimensionality, however, has not yet been attempted" should be revised since the references mentioned above have described systematic classifications of topological phases (the methods work in any dimension, although the results are only listed in 3d.)"

We thank the referee for pointing out that the formulation of our claims may have been too enthusiastic. We agree with the referee that there are published works that provide a full classification of topological insulators with some particular symmetries (two-fold symmetries being a well-known example). The central purpose of our work however, is to attempt the formulation of a fully general algorithm that can describe all types of crystal symmetry, and that includes both representation invariants and topological invariants not related to crystal symmetry.

We maintain that such a fully general classification of time-reversal symmetric topological insulators has not been attempted before. For example, we are not aware of any works that classify topological phases with D4 as well as time-reversal symmetry. We do agree that partial results in this direction, within restricted settings, have been made. For example, ref. 12 (Po et al.) provides a general classification of the representation theory side. We cite these earlier works and briefly discuss how we go beyond them in the revised manuscript.


The referee adds:
"In addition, at the conclusion of the paper, the authors write, "We cannot yet, however, give an explicit mathematical proof that these invariants exhaust all possible topological quantum numbers." This last sentence contradicts the claim of providing the first methodical algorithm. Thus, the authors must tone down their claims to correctly describe their results."

We thank the referee for pointing out that the presentation of these claims was confusing. We do not want to claim that our classification is complete, although it is certainly systematic. There is no way for us to prove that there may not be more topological invariants that are not accounted for in the present approach. We explicitly include this caveat in the revised manuscript, as suggested by the referee.


The referee points out:
"The authors define two phases to be topologically distinct "if smoothly deforming one into the other necessarily involves either closing the band gap around the Fermi level, or breaking a crystal symmetry." This definition is at odds with a classification from K-theory: two phases can be distinct by the above definition but equivalent in a K-theory classification if after adding a set of trivial bands, the two phases can be smoothly deformed into each other. Yet the authors say in the last line of the Discussion that their work could be checked by a comparison to K-theory. This contradiction is confusing: it is not clear whether the classification in the present work should, or should not, allow for the addition of trivial bands. This distinction was elucidated in Ref. 19 and has since been further explored in: Cano et al Phys. Rev. Lett. 120, 266401 (2018), Bouhon et al arXiv:1804.09719, and Bradlyn et al arXiv:1807.09729."

We agree with the referee that a discussion of this point should have been included in the original manuscript. In fact, our definition of smooth deformations may be trivially rephrased in a way that does allow the addition of what would be called trivial elements in K-theory. We chose not to mention this additional structure in the original manuscript because we aimed to provide as intuitive a picture as possible. Following the remarks of the referee, we now mention the possibility of adding trivial bands in the introduction, and include a brief description of how this can be done in the discussion section.


The referee then asks a question:
"The interplay between the torsion and representation invariants is not adequately discussed. Why, in the caption to Table 1, is the total classification the direct sum of the representation and torsion invariants?"

We thank the referee for pointing out that this was not clear in our original formulation. We include a brief discussion of this point in the revised manuscript.

In short, there are two ways of thinking about the direct sum. From a physics point of view, the total classification can be understood to be a direct sum because for any representation at a fixed point, there can alway be either a non-trivial or a trivial torsion invariant. The total classification is thus the sum of the independently allowed torsion and representation invariants.

Please notice that this does not mean there cannot be any relations between the torsion and representation invariants. In fact, such relations are already known in class A, and an example within class AII is presented in section C of our submitted manuscript. We realise that it may at first sight appear confusing that the classification is always a direct sum, in spite of the possible existence of relations between torsion and representation invariants, We therefore explicitly mention this aspect at the end of section C in the revised manuscript. In section V of the supplementary materials, there is a more detailed discussion of the relations between the different types of invariants.

A second way of understanding the appearance of a direct sum, is that mathematically, K-theory is known to split into a part describing the ordinary representation theory of the point group on the one hand, and a part addressing the time-reversal and other invariants on the other. The reason for this, is that in the computation, the result always comes in the form of an exact sequence, and since the K-theory of a point is always Z, the exact sequence always splits. This is why we can write the total classification as a direct sum of representation invariants and 'the rest', which we dub torsion invariants in the present manuscript. Notice again that this distinction is made on the level of the abelian groups, and that again, it does not exclude the possible existence of relations between the integer and Z2 invariants.


The referee next raises a crucial concern:
"The authors discuss the Chern number of a single band which has a degenerate point (Kramers partner) with another band (bottom left p3: "we can still consider the Chern number of just one band within each pair".) However, the Chern number of a band is not well defined unless it is separated by an energy gap from all other bands or has a symmetry eigenvalue (for example, if spin is conserved, then each band would correspond to the opposite spin.) Thus, without spin conservation, it is not correct to think of a nontrivial Z2 invariant as describing two bands with opposite Chern number, because the Chern number of each band individually is not well-defined.
Following this logic, I am not convinced that the vortex number of a single band which is degenerate with other bands is well defined: what happens if the Berry curvature is smeared over such a large radius that it reaches the degenerate high-symmetry points?"

We sincerely regret that the justification for being able to consider the two bands within a Kramers pair separately from one another was apparently not sufficiently clearly explained in the original submission, as it is central to our approach.

In fact, this result is not our own, and has been discussed in the literature in various guises. A rigorous treatment is given for example in the mathematical paper by De Nittis and Gomi (ref. 25 in the revised manuscript). That paper argues in detail why it is possible to view the ‘Kramers’ band structure as being built from two independent bands, and how the FKM invariant is related to the Chern number of just one of the two bands within a Kramers pair.

The main point is to realise that although two bands necessarily become degenerate at time-reversal symmetric point, this does not imply mixing of time-reversal partner states. That is, starting with a single non-degenerate states at k>0, parallel transport as prescribed by the (matrix-valued) Berry connection will result in a single non-degenerate state at k<0, rather than a superposition of two non-degenerate states. This special feature of the Kramers point is what allows us (and others before us) to smear Berry curvature across a Kramers point, as suggested by the referee, and still remain without any ambiguity as to which band contains the curvature.

We appreciate the referee pointing out that the justification for treating the Chern number within just one band of a Kramers pair as equivalent to the FKM invariant was not sufficiently clearly presented in the original manuscript. We also realised that the reference given above was hidden in the supplementary material of the original submission. In the revised version, we move this reference to the main text, and include additional explanation of this important issue.


The referee then asks for clarification:
"In addition, could the authors clarify whether all possible representation invariants are achievable? For example, in class AI with C2 symmetry, I believe it is impossible to have a single band with an odd number of C2 eigenvalues equal to -1 (for if it were possible, then the band would have an odd Chern number (see, i.e., Eq (24) of Ref. 6), which is incompatible with time-reversal symmetry.) Have the authors checked that in class AII, there are no forbidden representation invariants?"

We thank the referee for this interesting question. 

In the current manuscript, we focus exclusively on class AII, and are therefore in no position to judge whether or not the specific suggestion of the referee’s pertaining to class AI is correct or not. We do know that in class AII, there are some relations between the representations and the Chern number of a single band, which we discuss in detail in section V of the supplementary material.


The referee then points out some missing references:
"The original paper where Kane and Mele introduced the Z2 invariant for time-reversal symmetric topological insulators should be cited when referencing the Z2 invariant ((Kane and Mele, Phys. Rev. Lett. 95, 146802 (2005)).
Another glaring missing citation is a reference to Teo, Fu, Kane Phys. Rev. B 78, 045426 (2011), which introduced the mirror Chern number. This is one of the few crystalline topological invariants which has actually been observed in experiment (see Hsieh, et al Nature Communications volume 3, Article number: 982 (2012) (prediction) and Tanaka, et al Nature Physics volume 8, pages 800–803 (2012) (experiment))."

We thank the referee for pointing out these omissions, and include citations to these works in the revised manuscript.


The referee continues with the observation:
"The authors write, “In fact, it is easily seen that every combination of values for the two line invariants and one FKM invariant can be realised with precisely two distinct configurations of vortices on the high-symmetry points.” Can the authors elaborate on why this fact is “easily seen” (is the idea that each high-symmetry point with a vortex should not have a vortex in the other configuration, and vice versa?)."

We thank the referee for pointing out that this statement was perhaps presumptuous. We include a revised and more detailed statement in the revised manuscript.


The referee makes a suggestion:
"At the bottom right of p5, the authors write, “Looking at the allowed representations at Gamma, there is one real representation that allows for three vortices (or equivalently, a single charge-three vortex) to be formed there.” It would be helpful to include a table of allowed representations at Gamma in this case."

We thank the referee for the constructive feedback and the suggestion. However, we believe that the inclusion of a table as suggested by the referee would only add to the already large list of figures and tables in the manuscript without adding much in terms of content. Moreover, the representation theory of this group (and others) can nowadays be easily found on the Bilbao crystallographic server. We therefore prefer not to include such tables in the present manuscript.


The referee finishes with:
"At the bottom left of p6, the authors write, "For example, mirror symmetries or inversions force FKM3 to be trivial.” Yet, it is known that the FKM3 invariant is not trivial with inversion symmetry, since inversion eigenvalues can be used to compute the invariant (see Ref 23). This sentence, and similar sentences in the same section, must be corrected."

We thank the referee for pointing out this imprecision. We correct this and similar sentences in the revised version of the manuscript.

Response to the Second Referee

The referee starts out by stating:
"While the picture provided of Berry curvature vortices does indeed lead to the right predictions of invariants as can be confirmed by K-theory arguments in several cases, the argument presented here is not very rigorous. Since the occupied bands in an insulator can essentially be viewed as degenerate, the validity of an effective Hamiltonian approach for accounting for the Berry curvature at high symmetry points or elsewhere should be more adequately argued."

We thank the referee for the constructive remarks concerning this point. We agree with the referee that the work presented in the current manuscript does not include rigorous mathematical proofs. However, as mentioned before in the reply to the editor, the explicit aim of the present paper is to present a simple and intuitive picture for a topological classification of time-reversal invariant crystals that is accessible by anyone in the field, including experimental as well theoretical physicists. It is for this reason that we purposefully avoided formal mathematical expositions in the original submission.

To pursue the kind of mathematical rigour suggested by the referee, requires the computation of the K-theories associated with the topological phases we discuss in the manuscript, which would inevitably turn the paper into a formal mathematical treatise, rather than a physics article. We give some more formal arguments to back up our results in the supplementary material, but prefer not to put them in the main text.

Although we still feel that this is the correct choice for the present work, we appreciate the advice of the referee, and urged forward by their questions, two of us (JK and JdB) set out a collaboration with pure mathematicians to explicitly compute the K-theory groups involved in some of the cases where the current manuscript suggests new topological invariants. This work is now available as a purely mathematical paper on the ArXiv (https://arxiv.org/abs/1811.02592), and confirms the classification as proposed in the current manuscript in all cases for which we have explicit results in K-theory. Of course, this is still not a complete proof of our vortex picture, but it gives substantial evidence in that direction, and strengthens our confidence in the results of the present submission.


The referee continues by stating:
"The paper lists two classes of invariants, the representation invariants and torsion invariants. The latter are related to Berry curvature vortices and Chern numbers, and constraints on the changes in these at high symmetry and other points in the Brillouin zone are used to classify the allowed non trivial sets of Chern numbers. This approach while agreeing with those of K theory in many cases should be more adequately explained and argued.
(…)
While very likely correct, in its present form, its not very easy to judge the validity of the central technique of counting vortex configurations. The authors should try to make the exposition clearer. It may help to move the sections of the supplementary material in to the main text to make it more readable."


We regret that the referee did not find our exposition sufficiently clear.
Since the referee does not give concrete examples of where our explanation lacks clarity, or what they would consider to be sufficiently clear explanations, we cannot respond directly to this comment. However, we improved the overall clarity of the manuscript by adding additional explanations throughout the manuscript. As suggested by the referee, we also moved some references and explanations from the supplementary material to the main text. As pointed out also by the first referee, especially ref. 25 by De Nittis and Gomi is an important mathematical result that is central to the understanding of why the ‘Kramers’ band structure may be viewed as consisting of two separate bands, and how the FKM is related to the Chern number of just one of these two bands.


Finally, the referee remarks:
"In addition, the authors neglects to mention and compare their approach with that of earlier works which the other referee has listed."

We thank the referee for pointing out this omission, and include a comparison to and discussion of earlier works in the revised manuscript.