Topology in time-reversal symmetric crystals

The authors present a topological classification of insulators with crystalline symmetry in Class AII using a seemingly intuitive pictures of Berry curvature vortices.

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 a effective Hamiltonian approach for accounting for the Berry curvature at high symmetry points or elsewhere should be more adequately argued.

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.

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

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.

1. The authors present a classification of topological bands with crystal symmetry which is distinct from other works.

(These points are elaborated on in the report below.)
1. The authors overstate the novelty of their own work while failing to give proper credit to earlier papers. A comparison with earlier works is lacking.
2. At different points in the paper, different definitions of topological equivalence are used.
3. It is not clear how to combine the representation and torsion invariants.
4. It is not clear whether a vortex in a band which is degenerate at points with another band is well-defined.

The authors present a classification of topological bands with crystal symmetry, adapting the method they developed in Ref. 5 to the case with time-reversal symmetry. While the logic surrounding the representation invariants seems correct, the authors must give a stronger argument to show that the torsion invariants are well-defined and how they interact with the representation invariants. In addition, the authors make strong claims regarding the novelty of their work, while in fact existing classifications of topological bands with crystal symmetry have already been published; a comparison between their method and others is lacking. Thus, the following points must be addressed before the paper is suitable for publication:

1. 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.

Similarly, the claim, "A systematic classification of all possible topological phases in the presences 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.)

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.

2. 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.

3. 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?

4. 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?

5. 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?

In addition to the serious complaints addressed in my report, I have the following more minor comments:

1. 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)).

2. 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)).

3. 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?).

4. 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.

5. 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.