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Higher-order topological superconductors from Weyl semimetals

by Ammar Jahin , Apoorv Tiwari , Yuxuan Wang

Submission summary

As Contributors: Ammar Jahin
Preprint link: scipost_202103_00032v2
Date submitted: 2021-07-26 16:21
Submitted by: Jahin, Ammar
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


We propose that doped Weyl semimetals with time-reversal and certain crystalline symmetries are natural candidates to realize higher-order 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 finite-range attractive interaction favors a p + ip pairing symmetry. By analyzing its topological properties, we identify such a chiral pairing state as a higher-order topological superconductor, which depending on the existence of a four-fold roto-inversion 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 higher-order topology, including Wannier representability, Wannier spectrum, and defect classification approaches. For the R4z symmetric case, we provide a complete classification of the higher-order topological superconductors. We show that such second-order 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:
Editor-in-charge assigned

Author comments upon resubmission

The authors would like to thank the referees for their insightful comments that helped this work take a better shape.

List of changes

# List of Changes

- Move the discussion of appendix A to the main text
- Add definition of the $n$-cells.
- Add some discussion about how to generalize to more than $4$ Weyl points.
- Change the directions of the vector normal to the surfaces in Eq. (65).
- Add the discussion about how the Wannier obstruction directly leads to the chiral hinge mode.
- Make point about the requirement for long range interaction more clear.
- Add comparisons to previously found results.
- Modify the notation for $V_ {II^\prime}$.
- Fixed the factor of $2\pi$ between Berry flux and Chern number.
- Add what are the four components are under Eq. (20b).
- Properly reference Fig. 3 in the main text.
- Change the discussion about why $\mathsf T^2 = -1$ is inconsistent with our model.
- Mention that we take the Weyl points not to sit at the high-symmetry points.
- Change the introduction to include that we mainly consider spinless fermions.
- Fix typo above above Eq. (22) in the definition of the Green's function.
- Include a figure for the choice of the path $S^1_{\gamma}$.

Reports on this Submission

Report 2 by Rui-Xing Zhang on 2021-8-25 (Invited Report)


In both the resubmitted manuscript and the reply letter, the authors have done an excellent job in addressing the questions and concerns that I had in my previous report. In particular, they have significantly improved some figures and notations to greatly enhanced the readability of this work. Besides, they have clearly clarified the connection with previous works. Even though it remains unclear what real-world material system can serve as a platform to realize the proposed physics, we generally do not expect the first theory proposal to resolve all the issues that one might encounter. Now I think the comprehensiveness and novelty of this paper make it a beautiful theory work and SciPost-worthy on its own, and the material search will be a next-level future problem. Therefore, I am happy to recommend the current manuscript for publication in SciPost.

  • validity: top
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Report 1 by Frank Schindler on 2021-8-3 (Invited Report)


In their resubmission, the authors have addressed most of my requests and comments in a satisfactory manner. There is only one remaining issue:

In my first report, I had noted that:
"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."

The authors replied with:
"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."

This reply does not address my concern. Even when time-reversal symmetry squares to +1, so that there is no Kramers degeneracy, it still enforces a pairing of complex R4z eigenvalues. For this, we assume an eigenstate R4z |Ψ> = λ |Ψ>. Then, we find R4z T |Ψ> = T R4z |Ψ> = λ* T |Ψ>.

Given that Fig. 5 shows R4z eigenvalues that are not complex-conjugate to one another or real, the assumption that the commutator [T, R4z]=0 vanishes must be violated. This is unphysical -- all spatial symmetries should commute with time-reversal -- and needs to be amended before I can recommend publication of the manuscript.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Author:  Ammar Jahin  on 2021-09-02

(in reply to Report 1 by Frank Schindler on 2021-08-03)

We thank the referee for the further clarification, and indeed our previous response was not completely satisfactory.

The key question is that whether one should necessarily require $[T, R_{4z}]=0$. To this end, we would like first invoke a familiar conclusion — a two-fold symmetry necessarily commutes or anti-commutes with time-reversal symmetry. Indeed, in several well-known works on classifying topological crystalline insulators (TCI), e.g., the work by Shiozaki and Sato, the classification of a TCI with mirror symmetry M depends on whether $MT= TM$ or $ MT=-TM$.

In fact, here $MT=\pm TM$ are the only two options. This is due to the fact that $M, T$ and$ MT$ are two-fold symmetries. Since $MT$ is an anti-unitary symmetry, we can write $MT = U K$, where $U$ is a unitary operator and $K$ is complex conjugation. Then we have $(MT)^2 = UU^* = U(U^{-1})^T = \phi$. Since $MT$ is a two-fold symmetry, $\phi$ must be a diagonal operator. Therefore, $U = \phi U^T \phi$. Explicitly expressing the components, this is only possible if $\phi=\pm 1$. We thus have $MTMT=\pm 1$. Further assuming $M^2 =1$, and given that $T^2 = \pm 1$, we have $MT = \pm TM$.

However, to the best of our knowledge, no such constraints exist for a generic symmetry, including our four-fold symmetry $R_{4z}$. A counterexample is a generic spin rotation symmetry $S = \exp(i\sigma_z\theta/2)$, and a spinless time-reversal symmetry $T = K$. For a generic angle $\theta$, $ S$ and $T$ neither commute nor anti-commute. But this scenario is absolutely physical.

Going back to our case, time-reversal symmetry is broken in general but is restored at high-symmetry points. First, as we mentioned, the two $R_{4z}$ eigenvalues at every $K$ point in Fig. 5 are not TR partners. Instead each of them is invariant under TR. Second, because $R_{4z}$ does not commute or anti commute with T, its eigenvalues does not need to be real. In fact, our particular form of $R_{4z}$ in the BdG Hamiltonian comes from the fact that the $p+ip$ pairing order parameter carries an orbital angular momentum 1. In order to maintain rotation symmetry, a rotation in the pseudospin degree of freedom $\vec\tau=(\tau_x,\tau_y)$ needs to be incorporated into the rotation operator, i.e., $R_{4z} \propto \exp(i\tau_z \pi/4)$. This makes our case quite similar to the example with S above.

It is instructive to look at $R_{4z}^2=C_2$, which is a twofold symmetry. Its eigenstate $C_2 |Ψ\rangle = λ |Ψ\rangle$ satisfies the following relation: $ C_2 T |Ψ\rangle = \pm T C_2 |Ψ\rangle = \pm λ^* T |Ψ\rangle,$ where the $\pm$ depends on whether $C_2$ and $T$ commute or anti-commute. We see that if $T |Ψ\rangle = |Ψ\rangle$, then $λ$ needs to be either real (if $C_2$ and $T$ commute) or imaginary (if they anti-commute). In our case, $C_2 = i\tau_x$ and $T=K$, and we have $C_2 T = -T C_2$. Therefore $C_2$ eigenvalues are required to be imaginary. Indeed, by squaring the eigenvalues in Fig. 5, all $C_2$ eigenvalues are imaginary, consistent with the requirement of time-reversal symmetry.

We hope this addresses the referee’s insightful question.

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