SciPost Submission Page
Higherorder topological superconductors from Weyl semimetals
by Ammar Jahin , Apoorv Tiwari , Yuxuan Wang
This is not the latest submitted version.
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Ammar Jahin · Apoorv Tiwari 
Submission information  

Preprint Link:  scipost_202103_00032v1 (pdf) 
Date submitted:  20210330 16:48 
Submitted by:  Jahin, Ammar 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We propose that doped Weyl semimetals with four Weyl points are natural candidates to realize higherorder topological superconductors, which exhibit a fully gapped bulk while the surface hosts robust gapless chiral hinge states. We show that in such a doped Weyl semimetal, a featureless finiterange attractive interaction favors a p + ip pairing symmetry. By analyzing its topological properties, we identify such a chiral pairing state as a higherorder topological superconductor, which depending on the existence of a fourfold rotoinversion symmetry R4z, is either intrinsic (meaning that the corresponding hinge states can only be removed by closing the bulk gap, rather than modifying the surface states) or extrinsic. We achieve this understanding via various methods recently developed for higherorder topology, including Wannier representability, Wannier spectrum, and defect classification approaches. For the R4z symmetric case, we provide a complete classification of the higherorder topological superconductors. We show that such secondorder topological superconductors exhibit chiral hinge modes that are robust in the absence of interaction effects but can be eliminated at the cost of introducing surface topological order.
Current status:
Reports on this Submission
Report #2 by RuiXing Zhang (Referee 2) on 202166 (Invited Report)
 Cite as: RuiXing Zhang, Report on arXiv:scipost_202103_00032v1, delivered 20210606, doi: 10.21468/SciPost.Report.3030
Strengths
1. The emergence of intrinsic higherorder topological superconductivity from doped Weyl semimetal is an interesting idea/development in the field.
2. The results are rather comprehensive and the analysis is professional.
3. The paper is wellwritten.
Weaknesses
1. The applicability of this theory to nonminimal Weyl systems and realworld materials is unclear, possibly limiting the impact of this work.
2. Some statements are not well justified and could be overclaimed.
3. Missing some important references.
Report
In this manuscript, the authors introduced an interesting theory for emergent higherorder topological superconductivity (HOTSC) in doped R4zsymmetric Weyl semimetal (WSM). Starting from a minimal model for R4zcompatible WSM, the authors demonstrated that the most favorable pairing channel here is a nodeless oddparity one, directly leading to an R4zprotected HOTSC phase as a natural outcome. The authors also discussed a possible anomalous surface topological order that can symmetrically eliminate the R4zprotected higherorder topology in a symmetric & adiabatic way. When breaking R4z to C2, possible boundary obstructed extrinsic HOTSC phase is also discussed. As far as I can tell, the paper is wellwritten and the results in this work are interesting, professional, and comprehensive. However, the applicability of this theory to nonminimal Weyl systems and realworld materials is not quite clear to me. Besides, I also find some statements not very well justified and missing a highly relevant reference. Therefore, I can consider to recommend this work for publication in SciPost after the authors address following questions/comments.
1. It seems to me that chiral pwave being the most favorable pairing is a direct result of the dimension of the Hamiltonian matrix. Namely, a fourband Dirac model (including the BdG partners) apparently has much fewer choices of symmetryallowed mass terms than that of an eightband 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 twoband model? If yes or no, is the chiral pwave pairing still the leading instability? Can the same pairing always give rise to intrinsic HOTSC despite the number of Weyl points?
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?
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.
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 Nlayer 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?
5. Two comments on the Wannier obstruction discussed in Section 3.2.
(i) It has been known that being Wannier obstructed is a neither necessary nor sufficient condition for being higherorder (or firstorder) topological for BdG systems (see examples in Ref. 35 and 54). This is crucially different from the story in nonSC 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 higherorder 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 higherorder 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 bulkboundary 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.
(ii) Can the authors quickly check if the Wannier obstruction is stable or fragile?
6. Related to the above question of BBC, the definition of the defectbased second Chern number looks very interesting. Is it possible to relate this boundaryrelated invariant o the previous bulk symmetry indicator for Wannier obstruction? If this can be done, it might help explain the BBC.
Some minor comments:
7. The basis of V_II’ 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.
8. Slightly above Eq. 20, there is a typo in the definition of the Green function.
9. I find the Wannier spectrum calculation in Fig. 13 (b) of Appendix B quite useful to understand the bulk Wannier obstruction. So I would suggest move this figure to Section 3.2.
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.
11. At the beginning of Sec. 4, a definition for ncell is missing and this concept might not be familiar to general readers.
Report #1 by Frank Schindler (Referee 1) on 202161 (Invited Report)
 Cite as: Frank Schindler, Report on arXiv:scipost_202103_00032v1, delivered 20210601, doi: 10.21468/SciPost.Report.3005
Strengths
1 connects theory to physical realization
2 presents a full, selfcontained story
3 is fairly explicit/ comprehensible
Weaknesses
1 some claims are not generic/ require finetuning
2 classification results are not presented in context of extensive literature
Report
This manuscript comprehensively addresses a route to higherorder topological superconductors via longrange pairing in doped Weyl semimetals. It describes the normal state Hamiltonian, explicitly derives the BdG mean field theory, analyses its topological properties, and even treats the effects of surface topological order. Because physical realizations of higherorder superconductors are hard to come by, I believe that the paper is in principle worthy of publication. However, there are a couple of issues (detailed below) that need to be addressed before I can make such a recommendation.
Requested changes
1 Why should onsite/ shortrange pairing be disallowed? The case for higherorder topology could be argued much more compellingly if a mechanism favoring longrange pairing is given.
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.
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 momentumspace 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.
4 Fig 5 shows eigenvalues that are not compatible with TRS (they do not come in complexconjugated pairs). This can't be correct, because Eq. (35) is timereversal symmetric.
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)."
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 2π.
7 Figure 3 should be referenced and explained.
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?
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 π/4} with n an odd integer.
10 The sentence "A minimal model of such a system consists of two bands with four coplanar Weyl points." contains a hidden assumption (Weyl points not at highsymmetry points) that should be spelled out.
11 In the introduction it should be mentioned that the authors consider spinless electrons/ no SOC so that T^2 = +1.
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.
13 The sentence "Weyl semimetal with four Weyl nodes with attractive interaction naturally host a higherorder topological superconducting phase" is misleading because the authors show this only for longrange attractive interactions without arguing why these should be relevant in real materials.
Typos:
14 "NielsonNinomiya" on page 4
15 "the Green’s function G(k, ω) ≡ G(k, ω) = ..." on page 7
16 Eq. (28) has one minus sign too much
17 "Chiral" is capitalized two times on page 10
Author: Ammar Jahin on 20210726 [id 1614]
(in reply to Report 1 by Frank Schindler on 20210601)
We thank the referee for the useful and insightful comments.

Why should onsite/shortrange pairing be disallowed? The case for higherorder topology could be argued much more compellingly if a mechanism favoring longrange pairing is given.
 We do not argue that shortranged paring is disallowed in the pairing process. Rather, we show that as long as the pairing interaction is finiteranged, it naturally induces $p$wave pairing order with higherorder topology. We emphasize that this pairing mechanism does not necessarily require longrange interaction  our requirement only excludes e.g. onsite interaction, which is featureless in momentum space. In light of this comment, we have edited the text to make this point clearer.

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 higherorder superconductors. In contrast both references pointed out by the referee are formulated within the meanfield 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 highsymmetry points in the Brillouin zone. We have added such a discussion in the edited text.

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 momentumspace 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 timereversal 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.

Fig 5 shows eigenvalues that are not compatible with TRS (they do not come in complexconjugated pairs). This can't be correct, because Eq. (35) is timereversal symmetric.
 We believe this is a simple misunderstanding. The timereversal symmetry squares to $1$ and thus it does not impose Kramer degeneracy. Indeed the eigenvectors at these highsymmetry points are invariant under the action of timereversal symmetry.

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.

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.

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.

"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.

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.

The sentence "A minimal model of such a system consists of two bands with four coplanar Weyl points." contains a hidden assumption (Weyl points not at highsymmetry points) that should be spelled out.
 We agree, and have made this change in the revised version.

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.

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.

The sentence "Weyl semimetal with four Weyl nodes with attractive interaction naturally host a higherorder topological superconducting phase" is misleading because the authors show this only for longrange attractive interactions without arguing why these should be relevant in real materials.
 As we mentioned, our theory does not require longrange 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 higherorder topological superconudcting phase."
Author: Ammar Jahin on 20210726 [id 1615]
(in reply to Report 2 by RuiXing Zhang on 20210606)We thank the referee for the useful and insightful comments.
It seems to me that chiral pwave being the most favorable pairing is a direct result of the dimension of the Hamiltonian matrix. Namely, a fourband Dirac model (including the BdG partners) apparently has much fewer choices of symmetryallowed mass terms than that of an eightband 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 twoband model? If yes or no, is the chiral pwave pairing still the leading instability? Can the same pairing always give rise to intrinsic HOTSC despite the number of Weyl points?
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?
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.
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 Nlayer 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?
Two comments on the Wannier obstruction discussed in Section 3.2.
Related to the above question of BBC, the definition of the defectbased second Chern number looks very interesting. Is it possible to relate this boundaryrelated invariant o the previous bulk symmetry indicator for Wannier obstruction? If this can be done, it might help explain the BBC.
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.
Slightly above Eq. 20, there is a typo in the definition of the Green function.
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.
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.
At the beginning of Sec. 4, a definition for ncell is missing and this concept might not be familiar to general readers.