Higher-order topological superconductors from Weyl semimetals

We thank the referee for the useful and insightful comments.

1. It seems to me that chiral p-wave being the most favorable pairing is a direct result of the dimension of the Hamiltonian matrix. Namely, a four-band Dirac model (including the BdG partners) apparently has much fewer choices of symmetry-allowed mass terms than that of an eight-band model. This is probably why this specific pairing term is the only one that makes the SC state fully gapped, since the kinetic terms and the pairing terms can form a complete set of anticommuting 4 by 4 gamma matrices, as shown in Eq. 27. However, if the model has eight bands instead, I can imagine other pairing terms would be equally favorable in energy for being nodeless. This observation directly makes me concerned about the applicability of the present theory to a WSM with more than four Weyl nodes. Can a WSM with 4n Weyl points ($n>=1$) always be described by a two-band model? If yes or no, is the chiral p-wave pairing still the leading instability? Can the same pairing always give rise to intrinsic HOTSC despite the number of Weyl points?

   • This is an interesting question. While we focused on a minimal model for a Weyl semimetal with $\mathsf R_{4z}$ and time-reversal symmetries, our results can be straightforwardly generalized to the case with more Weyl points. As the referee hinted, there are two routes toward more Weyl points -- either by including additional copies of the two-band model, or by modifying the dispersion of the two-band model (for example via including longer-range hopping terms).
     • In the first case, the generalization is straightforward -- since the classification we obtained is $Z_2$, then the chiral superconducting system with $4n$ Weyl points has a nontrivial topology if $n$ is odd.
     • In the second case, one can repeat the same analysis for the modified two-band normal state, via the symmetry indicator approach or via the defect approach. It is not difficult to show that, again, the higher-order topology is nontrivial if $n$ is odd. We thank the referee for raising this point and we have added a discussion in the revised text, which we believe improves the quality of our work.

2. To further enhance the impact of this work, I believe a discussion on possible material candidates will be very helpful. Nevertheless, a basic requirement for the R4z WSM is its spinless nature. I wonder if the authors have any Weyl material candidate in mind that (i) is effective spinless; (ii) preserves some “effective” TRS. If not, can the authors discuss where and how to fulfill the above conditions simultaneously in an actual physical system?

   • An effective ''spinless" TRS can be realized by combining the physical TRS with certain spin rotation symmetries. Unfortunately, we are not immediately aware of a material realization. We are hopeful that with the recent development of topological quantum chemistry based on symmetry indicators, such candidate Weyl materials with $\mathsf{R} _{4z}$ symmetry can soon be identified. However, we believe this is out of the scope of the current work, and postpone such an investigation to future studies.

3. It is surprising that this work has missed an important relevant work on rotoinversion protected HOTI (Phys. Rev. B 98, 081110(R) by van Miert and Ortex). The authors might need to compare the current results with the nonsuperconducting counterpart in this PRB paper, especially a Z2 classification is also found in the HOTI case.

   • We thank the referee for pointing out this important reference. Although our methodology is very different, our results do agree with those in the reference. An important difference between our methodology and that of the work by van Miert and Ortex is that our approach is applicable to interacting as well as discordered/non-translational invariant systems that preseve the rotoinversion symmetry. Although we are interested in higher-order superconductors in the present work, our classification scheme can be readily adapted to insulators as well, by simply modifying the 2D blocks one uses in the cell decomposition described in Sec.4 of our paper. More precisely, the 2D blocks which host the $p+ip$ phase in the superconducting case, host Chern insulator phases in the insulating case. Upon doing so, we recover the $Z_2$ classification of van Miert and Ortex.

4. I am a little confused about the definition of the surface Chern number in Eq. 33. In particular, the authors defined “S^{+/-} is the set of upper/lower half of the layers”. Does this mean that for a N-layer slab, we label the layer collection of n=1,…,N/2 to be S^+ and that of n=N/2+1,…,N to be S^-? If so, I don’t quite get the relation in Eq. 34 as well. For example, I would expect C^+ {yz} = C^- {yz} =-0.5 due to the C2 symmetry. To make the hinge modes on the bottom xy surface to appear, I would require C^- {xy} = +0.5 = -C^+ {xy} to achieve the desired quantized change of surface Chern number. Am I correct?

   • Yes, this mentioned definition of $S^{+/-}$ is correct. We believe this was confusing because in the old version we had the direction normal to the surfaces to be fixed relative to the $x, y, z$-directions. We now adopted a more natural way of defining the normal vectors to be pointing outside of the sample. With such choice, everything is consistent with what is mentioned above. We thank the referee for point this out.

5. Two comments on the Wannier obstruction discussed in Section 3.2.

   1. It has been known that being Wannier obstructed is a neither necessary nor sufficient condition for being higher-order (or first-order) topological for BdG systems (see examples in Ref. 35 and 54). This is crucially different from the story in non-SC electronic insulators. Therefore, as far as I am concerned, proving the existence of a Wannier obstruction in a BdG system does NOT equal to the proof of higher-order topology, and vice versa. So the authors should be careful about making relevant claims. For example, the statement in Sec. 3.2 “…generally we have proven that an R4z Weyl semimetal with four Weyl nodes with attractive interaction naturally host a higher-order topological superconducting phase…” is NOT quite accurate to me, since only a proof of Wannier obstruction is shown in that section. To justify the above statement, the authors need to complete a proof or at least an argument of bulk-boundary correspondence (BBC) by showing that why such a Wannier obstruction will necessarily lead to the hinge modes. If such a proof of BBC is absent or difficult, I would suggest the authors modify the above claim accordingly.
      • We thank the referee for pointing this out. Indeed, in general, Wannier obstruction in the bulk is neither a sufficient nor a necessary condition for hinge modes. However, for our case, it is not difficult to provide a direct connection between the bulk obstruction and the chiral hinge modes. To this end, we note that both of the two-dimensional subsystems at $k_ z=0$ and $k_ z=\pi$ are Wannier representable. However, the Wyckoff centers of the Wannier states for the two subsystem are different -- it is at an obstructed position $(1/2,1/2)$ for $k_ z=0$ and at $(0,0)$ for $k_ z=\pi$. (For this reason there is no compatible Wannier representation for the full 3d system.) From our previous work (PRR, 043300 (2020)), this means that there is a corner zero mode state at $k_ z=0$, which disperses away at $k_ z =\pi$. Such a spectral flow at the hinge directly corresponds to the chiral hinge states. Inspired by the referee's comment, we have added such a discussion into the revised text.
   2. Can the authors quickly check if the Wannier obstruction is stable or fragile?
      • From our previous answer directly connecting the Wannier obstruction with the chiral hinge modes, it is stable.

6. Related to the above question of BBC, the definition of the defect-based second Chern number looks very interesting. Is it possible to relate this boundary-related invariant o the previous bulk symmetry indicator for Wannier obstruction? If this can be done, it might help explain the BBC.

   • We are not aware of any direct connection, although clearly they are obtained from the same set of information -- the pattern of the mass terms at high-symmetry points. Regarding the bulk-boundary correspondence, we believe our previous answer has clarified this point.

7. The basis of $VII’$ matrix in Eq. 17 is not clear to me. It would be better to relate the notation of I and I’ to that in Fig. 1.

   • We thank the referee for bringing this up to our attention. We have modifed the main text and hope it is clearer now.

8. Slightly above Eq. 20, there is a typo in the definition of the Green function.

   • We thank the referee for pointing out this typo.

9. I find the Wannier spectrum calculation in Fig. 13 (b) of Appendix B (now Appendix A in the new version) quite useful to understand the bulk Wannier obstruction. So I would suggest move this figure to Section 3.2.

   • We thank the referee for the interest. Appendix B and Fig. 13 (b) address a subtle difference between the Wannier spectrum and the surface energy spectrum in the presence of $\mathsf R_ {4z}$ symmetry -- even if the $xy$ surface is generally gapped, the Wannier spectrum in $(k_{x},k_{y})$ is gapless, reflecting the bulk Wannier obstruction. Heeding the referee's suggestion, we have made a reference to Fig. 13 (b) in the main text. However, the discussion on Fig. 13 is rather disconnected from the the main text. We hope our response above and the edited manuscript have made the bulk Wannier obstruction clear, and prefer to keep this figure in the Appendix.

10. About the definition of the second Chern number in 3.3.2, I would suggest visualizing the choice of hinge, $S_\gamma^1$, and other geometric quantities in a new schematic figure, which will definitely help explain the geometric meaning of the topological invariant.

    • We thank the referee for bringing this up to out attention. We have included such figure in the new version of the text.

11. At the beginning of Sec. 4, a definition for n-cell is missing and this concept might not be familiar to general readers.

    • We thank the referee for bringing this up to out attention. We have modified this in the new version of the text.

We thank the referee for the useful and insightful comments.

1. Why should on-site/short-range pairing be disallowed? The case for higher-order topology could be argued much more compellingly if a mechanism favoring long-range pairing is given.

   • We do not argue that short-ranged paring is disallowed in the pairing process. Rather, we show that as long as the pairing interaction is finite-ranged, it naturally induces $p$-wave pairing order with higher-order topology. We emphasize that this pairing mechanism does not necessarily require long-range interaction -- our requirement only excludes e.g. on-site interaction, which is featureless in momentum space. In light of this comment, we have edited the text to make this point clearer.

2. It should be clearly stated to what extent the classification results of Sec. 4 are original with respect to established references such as Science Advances eaaz8367 and Phys. Rev. Research 3 023086.

   • The classification derived in Sec.4 of our manuscript is applicable to but not limited to single particle/mean field descriptions of higher-order superconductors. In contrast both references pointed out by the referee are formulated within the mean-field BdG formalism. Another notable distinction is that our approach is based on a real space picture wherein we construct equivalence classes of ground state wavefunctions while the approach of the references is based on symmetry indicators constructed from symmetry eigenvalues at high-symmetry points in the Brillouin zone. We have added such a discussion in the edited text.

3. The Hamiltonian/symmetry discussion in 2.1 needs to connect to physical degrees of freedom. Right now, the Hamiltonian is stated, and then the representation of R4 symmetry is derived from it. This approach has two problems: the physical interpretation of the "internal band space" remains unclear, and R4 symmetry is not guaranteed. That is, the operator in Eq. (8) need not be a good R4 symmetry depending on the momentum-space structure of f(k) away from k=0. To ameliorate both problems, the authors should first state the microscopic orbitals, then derive the symmetry operations (T and R4) from them, and then derive the constraints these symmetries impose on the Hamiltonian.

   • We thank the referee for this suggestion. We believe this is a matter of taste. To reiterate our strategy, we consider a generic ''minimal model" for a time-reversal Weyl semimetal with $\mathsf R_{4z}$ symmetry, and we show that remarkably the form of symmetry operators can be completely fixed. The symmetry action on Wannier orbitals forming the Weyl semimetal can then be seen from the symmetry operators, such as that in Eq. (8). In our opinion, if we were constructing an effective Hamiltonian for a realistic material, the approach suggested by the Referee would have been more suitable. However in this work we analyze a generic model without a particular material in mind, we believe the approach we take is adequate.

4. Fig 5 shows eigenvalues that are not compatible with TRS (they do not come in complex-conjugated pairs). This can't be correct, because Eq. (35) is time-reversal symmetric.

   • We believe this is a simple misunderstanding. The time-reversal symmetry squares to $1$ and thus it does not impose Kramer degeneracy. Indeed the eigenvectors at these high-symmetry points are invariant under the action of time-reversal symmetry.

5. This claim on page 8 should be derived explicitly: "independent of the details of the band structure, ... the triplet channel ... is always an eigenmode of Eq. (23)."

   • This derivation is in Appendix A. In light of the potential interest to the readers, in the new manuscript we have moved it into the main text.

6. The quantity "Berry flux" should be defined before it is used. It is particularly confusing that the symbol "C" is used, suggesting a Chern number, but there seems to be a mismatch by a factor of $2\pi$.

   • We thank the referee for pointing this out and we have fixed it.

7. Figure 3 should be referenced and explained.

   • We thank the referee for pointing this out and we have properly referenced and discussed Figure 3 in the revised version.

8. "although here due to the lack of SU (2) symmetry in the band space, the four components are in general mixed": What are the "four components" here?

   • The four components referred to here are those of the paring order parameter, namely $(\Delta_ 0, d_ x, d_ y, d_z)$. We have modified the text to make this point clearer.

9. When the case with $T^2=-1$ is discussed on page 8, the use of R4 eigenvalues $\pm 1$ is inconsistent. $T^2=-1$ implies a spinful system where R4 has eigenvalues $e^{i n \pi/4}$ with n an odd integer.

   • We thank the refereeing for pointing this out. In the old version the argument against $\mathsf T^2=-1$ was made assuming the form of the $\mathsf R_ {4z}$ operator for the previous $\mathsf T^2=1$ case. This is clearly confusing. In the revised version, we adopted a more general argument showing $\mathsf T^2=-1$ is inconsistent with the $\mathsf R_ {4z}$ symmetry in the Weyl semimetal, which does not rely on the specific form of the $\mathsf R_{4z}$ operator and its eigenvalues.

10. The sentence "A minimal model of such a system consists of two bands with four co-planar Weyl points." contains a hidden assumption (Weyl points not at high-symmetry points) that should be spelled out.

    • We agree, and have made this change in the revised version.

11. In the introduction it should be mentioned that the authors consider spinless electrons/ no SOC so that $T^2 = +1$.

    • Indeed, even if we did briefly consider the $\mathsf T^2=-1$ case in Sec. 6.3, our main results are universally applicable to $\mathsf T^2=1$ cases only. We have followed the Referee's suggestion and made the change.

12. The sentence "a tube enclosing the hinge has a second Chern number protected by R4z symmetry" in the introduction is confusing because an individual hinge breaks R4z symmetry.

    • We thank the referee for raising this issue. The key point here is that the quantization of the defect topological invariant does not require spatial symmetries. Rather, spatial symmetries ensure the topological invariant is quantized to a nontrivial value. To be specific, although the tube does not preserve $\mathsf R_{4z}$ symmetry, as correctly pointed out by the referee, different points on the tube are related to one another by the action of the symmetry. Therefore choosing the mass to have a certain value (i.e. the mass vector to point in a certain arbitrary but fixed direction), automatically pins the value of the mass at certain other points on the tube. Furthermore the value of the masses deep in the bulk as well as outside are also fixed by symmetry requirements. Together these two considerations, pin the topological winding number of the mass texture on the tube. The winding number being directly related to the second Chern number consequently also enforces it, in this case to be quantized as an odd integer.

13. The sentence "Weyl semimetal with four Weyl nodes with attractive interaction naturally host a higher-order topological superconducting phase" is misleading because the authors show this only for long-range attractive interactions without arguing why these should be relevant in real materials.

    • As we mentioned, our theory does not require long-range interaction. However, we think that the referee's point is a valid point, and we have changed the sentence to "Weyl semimetal with four Weyl nodes with attractive interaction is a promising candidate to host a higher-order topological superconudcting phase."