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Higherorder topological superconductors from Weyl semimetals
by Ammar Jahin , Apoorv Tiwari , Yuxuan Wang
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Authors (as registered SciPost users):  Ammar Jahin · Apoorv Tiwari · Yuxuan Wang 
Submission information  

Preprint Link:  scipost_202103_00032v3 (pdf) 
Date submitted:  20211110 16:56 
Submitted by:  Jahin, Ammar 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We propose that doped Weyl semimetals with timereversal and certain crystalline symmetries 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.
List of changes
* Added more references.
* We addressed the issue raised in report 1 on the second version of the manuscript in a comment to the referee. We believe that no further changes are needed on that front.
Current status:
Reports on this Submission
Report #2 by Titus Neupert (Referee 6) on 20211116 (Invited Report)
 Cite as: Titus Neupert, Report on arXiv:scipost_202103_00032v3, delivered 20211116, doi: 10.21468/SciPost.Report.3855
Report
I have been asked by the editor to comment specifically on the issue of symmetry representations that has not been settled between authors and referee so far. I understand that there are two schools of thought about how to define symmetries in such tightbinding models. One strives to derive the symmetry representations from an orbital realization of a model (as the referee advocates), the other one simply posits symmetry representations (as the authors advocate). However, bridging between these two views is important and can be insightful.
Let me do it for the example of mirror symmetry brought up by the authors: A concrete realization could be a spinful chain of s orbitals that is ferromagnetically ordered and spinorbit coupled. The chain extends in x direction and the magnetization is also along x. Fundamental TRS $T$ ($T^2=1$) is broken and so is the mirror $M_y$ that sends $y\to y$ (the latter is broken due to the action of the mirror symmetry $\mathrm{i}\sigma_y$ on the polarized spins). However, the product $\tilde{T}=M_yT$ is a symmetry and along the 1D extent of the system it acts local in real space. Further $\tilde{T}^2=+1$, and $\tilde{T}M_x=M_x\tilde{T}$, where $M_x$ is the mirror symmetry along the chain. Hence, this is a situation in which the mirror $M_x$ and TRS (meaning the effective TRS $\tilde{T}$, i.e., some local in space antiunitary symmetry) anticommute instead of commute.
I think it would be helpful in connecting the two viewpoints mentioned above, if the authors could produce a similar physical motivation for the choice of symmetry representations that are used in their model, starting from spinful electrons in orbitals and the fundamental action of symmetries in the Lorentz group on Dirac electrons. After all, the authors are not deriving some more abstract problem like a classification table in which all sorts of representations may be listed, but they study a concrete model Hamiltonian with is meant to represent a physical system. To elucidate what type of microscopic system that is, I find such an analysis important and less a matter of taste.
Since TRS should be local and the system is 3D, I guess that arriving at it by a combination of a spatial symmetry with the fundamental TRS, as in my example above, is not an option. Probably assuming a collinear (anti)ferromagnet with negligible SOC, and combining the fundamental TRS with the spinoperator along that conserved spin direction, is more fruitful. If the authors encounter a fundamental obstacle in deriving the symmetry representation in this way, I think it would still be useful for the reader to explain why such a representation is not possible.
Report #1 by Frank Schindler (Referee 1) on 20211115 (Invited Report)
 Cite as: Frank Schindler, Report on arXiv:scipost_202103_00032v3, delivered 20211115, doi: 10.21468/SciPost.Report.3846
Report
After carefully considering the author's response, I remain unconvinced that my concern is satisfactorily addressed.
Let me recall that I had stated previously: "Given that Fig. 5 shows R4z eigenvalues that are not complexconjugate 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."
This inconsistency is still present in the current version of the manuscript.
In their reply, the authors note that in certain works in the classification literature, spatial symmetries are allowed to anticommute rather than commute with timereversal symmetry. This is a mathematical trick: If a spatial symmetry R anticommutes with T, then i*R commutes with T due to the antiunitary nature of time reversal. In this case, i*R is the physical symmetry.
Moreover, the authors allege that another counterexample to my statement is given by spin rotation symmetry $S = e^{i \sigma_z \theta/2}$, and spinless timereversal symmetry $T=K$. This is in fact no counterexample, but merely an inconsistent choice of symmetries. For spinful symmetries, spinful timereversal $T = i \sigma_y K$ should be used, which clearly commutes with spin rotations.
I therefore maintain my previous concern and cannot recommend publication of the manuscript in its present form.
Author: Ammar Jahin on 20211207 [id 2015]
(in reply to Report 1 by Frank Schindler on 20211115)In the latest reports, Referee Dr.~Frank Schindler reiterated that from a physical point of view, the rotoinversion operator $\mathsf{R}_{4z}$ and timereversal $\mathsf{T}$ must commute. According to the referee, while in certain examples, the commutation between the spatial symmetry operator and timereversal operator may be violated, it can be restored if the operators are chosen to their ``physical" forms. Referee Prof.~Titus Neupert very nicely summarized that the disagreement stemmed from two different conventions of defining symmetry operators, and asked us to elucidate the symmetry operations from the perspective of electronic orbitals.
We fully agree with both referees' point of view, and we thank them for helping us better understand their concerns. It turns out our initial result is fully consistent with the symmetry representation of orbital degrees of freedom, which can be clarified by a modification of notation.
Interpretation of rotoinversion symmetry in the normal state
Let us first consider the normal state on which timereversal is represented as $\mathsf T = K$, and and rotoinversion (which is a combination of $C_ {4z}$ and $M_ z$) operator is given by $\mathsf R_ {4z} = \hat f_ 1(0) \sigma_ x + \hat f_ 3(0) \sigma_ z$, where $\sigma$ represents our psudespin degree of freedom, and $\hat f^2_ 1(0) + \hat f^2_ 3(0) = 1$, as given in Eq. (8) in our paper. Importantly, here $\mathsf R_ {4z}$ and $ \mathsf{T}$ do commute.
Such a choice of symmetry operators can indeed be understood from the perspective of orbital degrees of freedom. Without loss of generality, let us consider the special case of $\hat f_ 1(0) = 0$, and $\hat f_ 3(0) = 1$, such that $\mathsf R_ {4z} = \sigma_ z$. In this case, we have two orbitals per unit cell both located at rotoinversion invariant points. The $\sigma_ z = 1\rangle$ orbital can be chosen to be , e.g., an $s$orbital, while the $\sigma_ z = 1\rangle$ can be a $d_ {xy}$orbital. This choice is not unique; for example $\sigma_ z = 1\rangle=d_ z^2\rangle$ and $\sigma_ z =  1\rangle = p_ z\rangle$ would also work.
For general values of $\hat f_ 1(0)$ and $\hat f_ 3(0)$, the physical interpretation of $\sigma_ z = \pm 1\rangle$ orbitals can be obtained as a linear superposition of $s\rangle$ and $d_ {xy}\rangle$ via a unitary transformation. Generally, the $\mathsf R_{4z} $ degree of freedom can be realized by, e.g., a (spinpolarized) $s$orbital or a $d_ {z^2}$orbital, while the $\mathsf R_ {4z} =1\rangle$ degree of freedom can be realized by, e.g. a $d_ {xy}$orbital or a $p_ z$orbital.
Noncommutation between rotoinversion symmetry and timereversal symmetry in the BdG Hamiltonian
Now we move on to the superconducting state with the $p+ip$ pairing term $H_ {\Delta} = \int_ {k} \Delta^{\alpha \beta} (k) c^\dagger_ {\alpha}(k) c^\dagger_ {\beta}( k)$, which transforms under rotoinversion in the nontrivial onedimensional representation as $\mathsf R_ {4z} : H_ {\Delta} \mapsto  i H_ {\Delta} $, since it carries angular momentum $1$. Consequently, the mean field Hamiltonian $H = H_ n + H_ {\Delta} + H^\dagger_ {\Delta}$, is not invariant under the action of operator $\mathsf R_ {4z} = \hat f_ 1(0) \sigma_ x + \hat f_ 3(0) \sigma_ z$.
We note that the additional factors of $i$ in front of the pairing terms that spoil the symmetry can be removed by an additional $U(1)$ transformation: $U(\frac{\pi}{4}) : c^\dagger_a (k) \rightarrow e^{i\frac{\pi}{4}} c^\dagger_a (k)$, and $U(\frac{\pi}{4}) : c_a (k) \rightarrow e^{i\frac{\pi}{4}} c_a (k)$. This $U(1)$ operation leaves the normal part of the Hamiltonian unchanged. Therefore, we define a new operator $\tilde R_ {4z} = U(\frac{\pi}{4}) \mathsf R_ {4z} $. (In the paper we had an abuse of notation in which we did not give this new combined symmetry a new symbol. We admit this is confusing, and have now fixed this.) We now have $[\tilde R_ {4z}, \mathsf T] \neq 0$ because of the added $U(1)$ factor. Crucially, even if $\tilde R_ {4z}$ is not the ``physical" rotoinversion operator, its action on the spatial part of the Hamiltonian is identical to rotoinversion. Indeed, we relied on the eigenvalues of $\tilde R_ {4z}$ to find the Wannier representation of the BdG Hamiltonian by treating it as that of an insulator.
In the Nambu space, this $U(1)$ operator is represented by $e^{i\frac{\pi}{4} \tau_ z}$, and thus we have $\tilde R_ {4z} = e^{i\frac{\pi}{4} \tau_ z} \mathsf R_ {4z} = e^{i\frac{\pi}{4} \tau_ z} (\hat f_ 1(0) \sigma_ x + \hat f_ 3(0) \sigma_ z) $, as given in the paper. We note that constructing the symmetry operator for the BdG Hamiltonian starting form the symmetry operator in the normal state and the onedimensional representation of the pairing terms in the way we just described follows closely the constuction given in Eqs. (7), (8), and (9) of [Trifunovic 2019] (https://arxiv.org/pdf/1910.11271.pdf).
We also note that this additional unitary operator in the Nambu space plays a similar role as $M_ y$ does in the example given by Referee Titus Neupert in obtaining $[M_ x,\tilde T]\neq 0$, although there the modification was on the timereversal operator $\tilde T \equiv M_ y T$.
Finally, we also note that our choice of timereversal symmetry $\mathsf{T}=K$ is, strictly speaking, also not ``physical", since the physical timereversal symmetry operator squares to $1$. As is well understood and similar to what was mentioned by Referee Neupert, here $\mathsf{T}=K$ can be understood as a composite of the physical timereversal symmetry $\mathsf{T}' = is_ y K$ and a spin rotation $is_y$. In this context, we assume the band electrons we consider is fully spin polarized.