A Hamiltonian Approach for Obtaining Irreducible Projective Representations and the k\cdot p Perturbation for Anti-unitary Symmetry Groups
aa r X i v : . [ m a t h - ph ] J a n A Hamiltonian Approach for Obtaining Irreducible Projective Representations andthe k · p Perturbation for Anti-unitary Symmetry Groups
Zhen-Yuan Yang, Jian Yang, Chen Fang, and Zheng-Xin Liu ∗ Department of physics, Renmin University, Beijing 100876, China. Beijing National Research Center for Condensed Matter Physics,and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China (Dated: January 21, 2021)As is known, the irreducible projective representations (Reps) of anti-unitary groups contain threedifferent situations, namely, the real, the complex and quaternion types with torsion number 1,2,4respectively. This subtlety increases the complexity in obtaining irreducible projective Reps of anti-unitary groups. In the present work, a physical approach is introduced to derive the condition ofirreducibility for projective Reps of anti-unitary groups. Then a practical procedure is providedto reduce an arbitrary projective Rep into direct sum of irreducible ones. The central idea is toconstruct a hermitian Hamiltonian matrix which commutes with the representation of every groupelement g ∈ G , such that each of its eigenspaces forms an irreducible representation space of thegroup G . Thus the Rep is completely reduced in the eigenspaces of the Hamiltonian. This approachis applied in the k · p effective theory at the high symmetry points (HSPs) of the Brillouin zone forquasi-particle excitations in magnetic materials. After giving the criterion to judge the power ofsingle-particle dispersion around a HSP, we then provide a systematic procedure to construct the k · p effective model. PACS numbers:
I. INTRODUCTION
Irreducible projective representations (IPReps) ofgroups, including the irreducible linear Reps as the triv-ial class of IPReps, play important roles in physics[1–7].In condensed matter physics, IPReps for discrete groupsare widely used in obtaining selection rules or analyzingspectrum degeneracy[8]. For instance, in the band theoryof itinerant electrons hopping in a crystal, the symmetrygroup is a space group whence the degeneracy of the en-ergy spectrum at a momentum point is determined bythe dimensions of IPReps of the little co-group[9].Owing to the importance of IPReps, it is urgent tojudge if a Rep is reducible or not. For a finite uni-tary group H , a (projective) Rep D ( H ) is irreducible ifit satisfies the following condition, | H | P h ∈ H | χ h | = 1 , where χ ( ν ) ( h ) = Tr D ( ν ) ( h ) is the character of the element h ∈ H . When D ( H ) is reducible, then | H | P h ∈ H | χ h | = P ν a ν , where a ν is the multiplicity of the irreducible Rep( ν ) contained in D ( H ). In this case, we need to trans-form it into a direct sum of irreducible Reps. The eigen-function method[9] is an efficient way of performing thisreduction.On the other hand, anti-unitary groups attract moreand more interests. The well known Kramers degen-eracy is a consequence of time-reversal symmetry forfermions with half-odd-integer spin. Time reversal alsoprotects the helical gapless edge modes in topological ∗ Electronic address: [email protected] insulators[10, 11] or topological superconductors[12, 13].Especially, a large amount of materials in nature ex-hibit magnetic long-range order, the symmetries for someof these materials are described by anti-unitary groupscalled the magnetic space groups[14], where the anti-unitary operations are generally combination of time re-versal operation T and certain unitary space-group el-ement. The irreducible Reps (also called co-Reps) ofthe magnetic space groups are helpful to understand theproperties of these materials. Especially, the low-energyphysics of the quasi-particles at high symmetry points(HSPs) of the Brillouin zone (BZ) are characterized bythe irreducible projective Reps of the little co-groups.For anti-unitary groups, there are three types of ir-reducible Reps which are characterized by the torsionnumber. Supposing that M ( G ) is an irreducible Repof an anti-unitary group G , and H is the halving uni-tary subgroup H ⊂ G with G = H + T H ( T isanti-unitary). Then the torsion number is given by R = | H | P h ∈ H | χ ( h ) | , where χ ( h ) = Tr M ( h ) is thecharacter of h . If R = 1, the irreducible Rep M ( G ) be-longs to the real type; if R = 2, then M ( G ) belongsto the complex type; if R = 4 then M ( G ) belongs tothe quaternion type[15]. This subtlety of anti-unitarygroups increases the complexity in reducing an arbitraryprojective Rep into the direct sum of irreducible ones,especially if some IPReps appea multipole times in thereucible Rep.In the present paper, from a physical approach we de-rive the criterion to judge the irreducibility[16] of a pro-jective Rep M ( G ) for a finite anti-unitary group G ,1 | H | X h ∈ H
12 [ χ ( h ) χ ∗ ( h ) + Tr[ M ( T h ) M ∗ ( T h )]] = 1 , or equivalently | H | X h ∈ H (cid:2) χ ( h ) χ ∗ ( h ) + ω ( T h, T h ) χ (( T h ) ) (cid:3) = 1 , (1) where ω ( T h, T h ) is the factor system of the projectiveRep. In this approach, we consider Hermitian Hamilto-nians in terms of single-particle bilinear operators whichare commuting with all of the symmetry operations in G .If the only existing Hamiltonian is proportional to theidentity matrix, then the Rep M ( G ) is irreducible. Oth-erwise, if there exist other linearly independent Hamil-tonian, then M ( G ) is reducible and the energies of theHamiltonian can be used to distinguished each of the ir-reducible subspace. This provides an efficient method toreduce an arbitrary reducible Rep into a direct sum of ir-reducible ones. The advantage of the method is that noinformation of the irreducible Reps of the groups needto be known beforehand. We further generalize this ap-proach to judge the power of the quasi-particle disper-sions in magnetic semimetals, and then to obtain the k · p effective models [17, 18] at the HSPs in the BZ.The rest of the paper is organized as follows. In sec-tion II, we worm up by reviewing the IPReps of uni-tary groups, and then derive the formula (1) for anti-unitary groups and interpret it in a physical Hamiltonianapproach. In section III, applying the Hamiltonian ap-proach we provide the procedure to reduce an arbitraryRep of finite groups (either unitary or anti-unitary) intoa direct sum of IPReps. In section IV, we provide thecriterion to judge if the degeneracy protected by IPRepsof anti-unitary groups can be lift by certain perturba-tions or not, and then give the method to construct k · p effective Hamiltonian for magnetic materials. Section Vis devoted to the conclusions and discussions.Since any Rep of a finite group (no matter unitary oranti-unitary) can be transformed into a unitary one, inthe present work we only discuss unitary Reps. II. A HAMILTONIAN APPROACH:CONDITION FOR IRREDUCIBLE PROJECTIVEREPSA. Unitary Groups
Since the character of the identity Rep ( I ) is χ ( I ) ( h ) =1 for any h ∈ H , the following quantity a ( ν × ν ∗ )( I ) = 1 | H | X h ∈ H | χ ( ν ) h | (cid:16) χ ( I ) ( h ) (cid:17) ∗ = 1 | H | X h ∈ H Tr[ D ( ν ) ( h ) ⊗ D ( ν ) ∗ ( h )] , stands for the multiplicity of the identity Rep appearingin the reduced Rep of the direct product ( ν × ν ∗ ), where( ν ) ∗ is the complex conjugate of ( ν ). Then the conditionof irreducibility of ( ν ) can be interpreted as the following:the direct product ( ν ) × ( ν ) ∗ contains only one identityRep, namely a ( ν × ν ∗ )( I ) = 1.The expression a ( ν × ν ∗ )( I ) = 1 has a physical interpre-tation. Suppose the identical particle ψ † has d inter-nal components ψ † = ( ψ † , ψ † , ..., ψ † d ), which carries anRep ( ν ) of the symmetry group H . This means thatˆ hψ † i ˆ h − = P j D ( ν ) ji ( h ) ψ † j or equivalentlyˆ hψ † ˆ h − = ψ † D ( ν ) ( h ) . The hermitian conjugation gives ˆ hψ ˆ h − = [ D ( ν ) ( h )] † ψ. The energy spectrum is described by the single-particleHamiltonian ˆ H = X i ψ † i Γ ij ψ j = ψ † Γ ψ, (2)where Γ is an d × d matrix. The symmetry group H means that the Hamiltonian is invariant under all thesymmetry operations in the group H . In other words, forany h ∈ H , we have ˆ h ˆ H ˆ h − = ˆ H , which is equivalentto D ( ν ) ( h )Γ[ D ( ν ) ( h )] † = Γ . (3)Schur’s lemma indicates that when ( ν ) is irreducible,then Γ must be proportional to the identity matrixΓ ∝ I . If there exist another linearly independent ma-trix Γ satisfying (3), then it must have at least two eigen-values. The eigenspace of each eigenvalue is closed underaction of H and hence form a Rep space of H . Thismeans that the Rep ( ν ) is reducible. Therefore, if I isthe only one linearly independent matrix satisfying (3),then the d -fold degenerate energy level of H cannot belift and consequently ( ν ) is irreducible.The equation (3) can be expanded in the following form X j,k D ( ν ) ij ( h )Γ jk D ( ν ) ∗ lk ( h ) = X j,k (cid:16) D ( ν ) ( h ) ⊗ D ( ν ) ∗ ( h ) (cid:17) il,jk Γ jk = Γ il for all h ∈ H . If we reshape the matrix Γ into an d -component column vector (if the matrix Γ is reshapedinto the d -component vector column by column, then itshould be transposed into Γ T before the reshaping), thenthis vector is the eigenvector of D ( ν ) ( h ) ⊗ D ( ν ) ∗ ( h ) witheigenvalue 1 for all h ∈ H , i.e. it carries the identity Repof H . In other words, the vector Γ is the CG coefficient[19, 20] that combines the bases of ( ν ) and ( ν ) ∗ to airreducible basis that belongs to the identity Rep χ ( I ) ( h ) =1 . If ( ν ) is irreducible, then the CG coefficient is unique. Above discussion is valid no matter the Rep ( ν ) is lin-ear or projective. B. Anti-unitary groups
In the following we generalize above approach to anti-unitary groups. Consider an anti-unitary group G with G = H + T H , where H ∈ G is the halving unitarysubgroup and T is an anti-unitary element of the lowestorder.If G is of type-I[21], namely, T = E , then G is eithera direct product group G = H × Z T or a semi-directproduct G = H ⋊ Z T , where Z T = { E, T } . If T ∈ G (here T is the time-reversal operation which commuteswith all the other elements), then we choose T = T ;otherwise, T = uT , where u / ∈ G is a unitary operationsatisfying T = u = E .On the other hand, if G is of type-II, then T n = E with n ≥
2, hence G cannot be written in forms of directproduct or semi-direct product of a unitary group with Z T . Obviously, the order of T is at least 4 and T ≡ σ is a unitary element in H , σ ∈ H .We consider an d -dimensional unitary projective Repof G . Any element g ∈ G is represented as ˆ g = M ( g ) K s ( g ) , which satisfies the relations M † ( g ) M ( g ) = I and M ( g ) K s ( g ) M ( g ) K s ( g ) = ω ( g , g ) M ( g g ) K s ( g g ) , where s ( g ) = 1, K s ( g ) = K if g is anti-unitary and s ( g ) =0, K s ( g ) = I otherwise. The factor system ω ( g , g )satisfies the cocycle equation ω s ( g ) ( g , g ) ω − ( g g , g ) ω ( g , g g ) ω − ( g , g ) = 1 . Now we derive the condition for the irreducibility of M ( g ) K s ( g ) .
1. General Discussion
Since unitary group elements are easier to handle, weexpect that the irreducibility can be judged from the re-strict Rep of the subgroup H . Noticing that M ( H ) ispossibly reducible even if M ( G ) is irreducible, we have1 | H | X h ∈ H Tr[ M ( h ) ⊗ M ∗ ( h )] ≥ . Actually P ( I ) = | H | P h ∈ H M ( h ) ⊗ M ∗ ( h ) is the projectoronto the subspace of identity Reps contained in the directproduct Rep M ( h ) ⊗ M ∗ ( h ). The eigenvalues of P ( I ) areeither 1 (which occurs at least once) or 0, hence Tr P ( I ) ≥
1. We need to find a way to include the restrictions fromthe anti-unitary group elements. Adopting the physi-cal argument as discussed in Sec. II A, we consider a d -component particle ψ † which carries the (co-)Rep of g ∈ G , ˆ gψ † ˆ g − = ψ † M ( g ) K s ( g ) . The Hamiltonian takes the same form of (2), which isinvariant under the action of all the group elements,ˆ g ˆ H ˆ g − = ˆ H , namely, M ( h )Γ M ( h ) † = Γ , h ∈ H (4) M ( T )Γ ∗ M ( T ) † = Γ . (5)Similar to the discussion for unitary groups, the Γ matrixcan be considered the CG coefficient that combines theproduct Rep M ( g ) ⊗ M ∗ ( g ) K s ( g ) , g ∈ G (a linear Rep)into the identity Rep. Since the identity matrix obvi-ously satisfies the above two equations, the product Repcontains at least one identity Rep. We expect that theidentity matrix is the unique linearly independent ma-trix satisfying (4) and (5) if the Rep M ( g ) K s ( g ) , g ∈ G isirreducible.However, above statement is too strong for anti-unitary groups. We need one more constraint for Γ. No-tice that if a matrix commutes with an irreducible (pro-jective) Rep of an anti-unitary group, then this matrixmay have two eigenvalues which are mutually complexconjugate to each other [21]. To generalize the Schur’slemma to anti-unitary groups, the matrix Γ needs to beHermitian. Namely, if an Hermitian matrix commuteswith the irreducible projective Reps of all the group el-ements of an anti-unitary group, then this matrix mustbe proportional to the identity matrix.Hence, in addition to (4) and (5), we should furtherrequire that Γ † = Γ . (6)If a non-hermitian matrix Γ satisfies (4) and (5), then ob-viously its hermitian conjugate Γ † also does. Therefore,the linear combination (Γ + Γ † ) is the required hermitianmatrix [43].Therefore, when making using of the characters ofthe unitary subgroup H to judge the irreducibility of M ( g ) K s ( g ) , g ∈ G , we need a projection operator P HT = P H P T to project onto the subspace formed by hermitianand T symmetric matrices. P HT is equivalent to projectonto the eigenvectors of M ( T ) ⊗ M ∗ ( T ) K with eigen-value 1 with the condition that the matrix form of theseeigenvectors are hermitian.Therefore, considering (4), (5) and (6), the irreducibil-ity requires thatTr( P ( I ) P HT ) = 1 | H | X h ∈ H Tr[ M ( h ) ⊗ M ∗ ( h ) P HT ] = 1 , (7)namely, when projecting onto the hermitian and T sym-metric subspace, the identity Rep only appears once inthe direct product Rep M ( H ) ⊗ M ∗ ( H ).Eq. (7) is a general expression of the criterion thata Rep of anti-unitary groups should meet if it is irre-ducible. However, the construction of the projection op-erator P HT is not straightforward. In the following wefirst consider a relatively simple case, i.e. the type-I anti-unitary groups, and then generalize the conclusion to ar-bitrary anti-unitary groups.
2. type-I anti-unitary groups
For type-I anti-unitary groups with T ≡ σ = E , sit-uations are much simpler. For a unitary Rep, we have M † ( T ) M ( T ) = I . On the other hand, T = E indicates[ M ( T ) K ] = M ( T ) M ∗ ( T ) = η ≡ ω ( T , T ) , where η = ± T = M ∗ ( T )Γ M T ( T ), namely M ( T )Γ T = M ( T ) M ∗ ( T )Γ M T ( T ) = η Γ M T ( T ) . Defining ˜Γ = Γ M T ( T ) , then we have˜Γ T = η ˜Γ . (8)This means that ˜Γ is either symmetric (if η = 1) or anti-symmetric (if η = − η -symmetric if it satisfies (8).Since Γ = ˜Γ M ∗ ( T ), we rewrite the Hamiltonian asˆ H = ψ † ˜Γ M ∗ ( T ) ψ = ψ † ˜Γ ˜ ψ, then the basis ψ undergoes a unitary transformation ψ → ˜ ψ = M ∗ ( T ) ψ . Under the action of h ∈ H , ˜ ψ varyas ˆ h ˜ ψ ˆ h − = M ∗ ( T ) M † ( h ) ψ = M ∗ ( T ) M † ( h ) M T ( T ) ˜ ψ .For convenience, we define the following Rep for h ∈ H , F ( h ) = M ( T ) M ∗ ( h ) M † ( T ) , (9)which is equivalent to M ∗ ( h ) with Tr F ( h ) =Tr M ∗ ( h ) = χ ∗ ( h ). Accordingly, ˜ ψ vary as ˆ h ˜ ψ ˆ h − = F T ( h ) ˜ ψ . Hence, the condition ˆ h ˆ H ˆ h − = ˆ H requiresthat M ( h )˜Γ F T ( h ) = ˜Γ , (10)which is the deformation of (4). Similarly, (5) is trans-formed into M ( T )˜Γ ∗ M T ( T ) = ˜Γ . (11)As before, ˜Γ can be considered as the CG coefficientthat couples the direct product Rep V ( g ) = M ( g ) ⊗ F ( g ) K s ( g ) , g ∈ G (a linear Rep) to the identity Rep,namely X ij V kl,ij ( g ) K s ( g ) ˜Γ ij = X ij M ki ( g ) F lj ( g ) K s ( g ) ˜Γ ij = ˜Γ kl K s ( g ) , (12)with V ( T ) = M ( T ) ⊗ M ( T ). If the CG coefficientmatrix ˜Γ satisfies the η -symmetry condition (8), thenwe only need to consider the unitary elements h ∈ H . Obviously, ˜Γ = M T ( T ) ( i.e. Γ = I ) satisfies the rela-tions (8) and (10). The irreducibility of M ( g ) K s ( g ) , g ∈ G indicates that there is a unique linearly independent so-lution. In other words, when projected onto the η -symmetric subspace by the projection operator P η , theproduct Rep V ( h ) = M ( h ) ⊗ F ( h ) only contains a singleidentity Rep, a [ M ⊗ F ] η ( I ) = 1 | H | X h ∈ H Tr [ M ( h ) ⊗ F ( h ) P η ] = 1 . (13)To obtain the matrix form of [ M ( h ) ⊗ F ( h ) P η ], wedevide M ( h ) ⊗ F ( h ) into two parts, X ij (cid:0) M ( h ) ⊗ F ( h ) (cid:1) kl,ij ˜Γ ij = X ij (cid:2) M ki ( h ) F lj ( h ) + η M kj ( h ) F li ( h ) (cid:3) ˜Γ ij + X ij (cid:2) M ki ( h ) F lj ( h ) − η M kj ( h ) F li ( h ) (cid:3) ˜Γ ij . Noticing that the second summation on the righthandside vanishes owing to ˜Γ ij = η ˜Γ ji , so we have[ V η ( h )] kl,ij = 12 (cid:0) M ki ( h ) F lj ( h ) + η M kj ( h ) F li ( h ) (cid:1) , (14)where we have used the notation V η ( h ) = [ M ( h ) ⊗ F ( h ) P η ].Introducing the unit twist matrix( T ) kl,ij = δ kj δ li with ( X T ) kl,ij = X kl,ji for an arbitrary matrix X , thenthe projection operator P η can be expressed as P η = 12 ( I + T η ) , where I is the identity matrix and T η = η T . Hence V η ( h ) in (14) can be written as V η ( h ) = (cid:0) M ( h ) ⊗ F ( h ) (cid:1) P η = 12 V ( h )( I + T η ) . Although V η ( h ) does not form a Rep of H , the com-mon eigenvector of V η ( h ) with eigenvalue 1 does carriesthe identity Rep of H (see Theorem 1 for the special casein which σ = E ). Defining P ( I ) = | H | P h ∈ H V ( h ), abovestatement indicates that if ˜Γ satisfies P ( I ) P η ˜Γ ij = ˜Γ kl , (15)then it simultaneously satisfies the relations (8) and (10).Furthermore, by choosing proper bases in the supportingspace of P ( I ) P η , the T -symmetry condition (11) andfinally the hemiticity condition (6) can be ensured (seeAppendix A for details). Thus the criterion (13) for theirreducibility is valid.From the matrix form in (14), the criterion (13) canbe expressed in terms of the characters χ ( h ) = Tr M ( h )of the unitary elements h ∈ H , namely1 | H | X h ∈ H (cid:16) χ ( h ) χ ∗ ( h ) + η Tr [ F ( h ) M ( h )] (cid:17) = 1 , where Tr F ( h ) = χ ∗ ( h ) has been used. Furthermore, bydenoting ¯ h = T − hT = T hT , we have F ( h ) = M ( T ) M ∗ ( h ) M † ( T )= ω ( T , h ) ω ( T h, T ) ω ( T , T ) M (¯ h ) . Above can be further simplified using the cocycle rela-tion ω − ( T , h ) ω − (¯ h, h ) ω ( T h, T h ) ω − ( T h, T ) = 1,which yields ω ( T , h ) ω ( T h, T ) = ω ( T h,T h ) ω (¯ h,h ) . Notic-ing that η = 1, therefore we have η Tr [ F ( h ) M ( h )] = ω ( T h, T h ) ω (¯ h, h ) Tr [ M (¯ h ) M ( h )]= ω ( T h, T h )Tr [ M (¯ hh )]= ω ( T h, T h ) χ (( T h ) )= Tr [ M ( T h ) M ∗ ( T h )] . Finally, we reach the simplified irreducible condition12 | H | X h ∈ H (cid:16) χ ( h ) χ ∗ ( h ) + ω ( T h, T h ) χ (cid:0) ( T h ) (cid:1)(cid:17) = 1 . (16)Above expression is independent on the gauge choice ofthe projective Rep. The factor system ω ( T h, T h ) canbe avoided by the replacement ω ( T h, T h ) χ (cid:0) ( T h ) (cid:1) =Tr [ M ( T h ) M ∗ ( T h )].In the following we show that above condition of irre-ducibility also works for type-II anti-unitary groups.
3. type-II anti-unitary groups
For type-II anti-unitary groups, we denote T ≡ σ .Similar to previous discussion, if we define ˜Γ = Γ M T ( T ),then (10) and (11) are the conditions ˜Γ should satisfy.Furthermore, recalling Γ is hermitian, we have˜Γ T = M ( T )Γ T = M ( T ) M ∗ ( T )Γ M T ( T )= η M ( σ )˜Γ . (17)The self-consistency condition (˜Γ T ) T = ˜Γ requires that˜Γ = M ( σ )˜Γ F T ( σ ) with F ( σ ) = η M ( σ ), or equivalently V ( σ )˜Γ = M ( σ ) ⊗ F ( σ )˜Γ = ˜Γ . This requirement is guaranteed if (10) is satisfied. Therefore, the T symmetry condition (5) and the her-miticity condition (6) combine to a single restriction (17),i.e. ˜Γ = [ η M ( σ )˜Γ] T . Now we define a generalized twistoperator T η which transforms ˜Γ into [ η M ( σ )˜Γ] T , T η = η T [ M ( σ ) ⊗ I ] . (18)It holds that ( T η ) = M ( σ ) ⊗ F ( σ ) = V ( σ ). Thus T η defines a generalized ‘transpose’ of ˜Γ given that V ( σ )˜Γ =˜Γ is satisfied.When projected to the eigenspace of V ( σ ) with eigen-value 1, the operator T η has eigenvalues ±
1. Thereforethe projector onto the generalized η -symmetric subspaceis given by P η = 12 P σ ( I + T η ) , where P σ is a projection onto the eigenspace of V ( σ ) witheigenvalue 1. Defining the projector onto the subspaceof identity Reps as P ( I ) = | H | P h ∈ H V ( h ), then it isobvious that P ( I ) P σ = P ( I ) . Hence the irreducibilitycondition (13) can be written as a [ M ⊗ F ] η ( I ) = Tr( P ( I ) P η ) = 12 Tr (cid:0) P ( I ) ( I + T η ) (cid:1) = 1 . (19)After some calculations, above criterion of irreducibilitycan be simplified to the same form as (16) (see AppendixB).From the definition of torsion number of irreducibleReps and the equation (16), one can easily verify thefollowing relation for any anti-unitary group G = H + T H ,1 | H | X h ∈ H ω ( T h, T h ) χ (cid:0) ( T h ) (cid:1) = 1 | H | X u ∈ T H Tr[ M ( u ) M ∗ u )]= , if R = 10 , if R = 2 − , if R = 4 , which provides another way to obtain the torsion num-ber. III. HAMILTONIAN APPROACH FOR THEREDUCTION OF PROJECTIVE REPS
The criterion of judging the irreducibility actuallyprovides a practical procedure to reduce reducible pro-jective Reps of finite groups. In the following, we discussunitary groups and anti-unitary groups separately.
A. Reduction of Reps for Unitary groups
For a general hermitian Hamiltonian matrix Γ satisfy-ing (3), each of its eigenspace is an irreducible subspaceof the unitary group H . Namely, the eigenvalues of Γcan be used to label the irreducible projective Reps of H . In order to simultaneously block diagonalize the re-strict Rep of H and its subgroups, we can make use ofthe class operators of H and those of its subgroups[9] tolift the degeneracy of Γ.Therefore, the central step is to construct the hermi-tian Hamiltonian matrix Γ. Here we summarize the re-duction procedure in the following three steps:(1) Obtain the subspace L ( I ) which carries the iden-tity Reps of M ( H ) ⊗ M ∗ ( H ), namely, find all the bases v ( I ) i ∈ L ( I ) such that for any group element h ∈ H , M ( h ) ⊗ M ∗ ( h ) v ( I ) i = v ( I ) i ;(2) Chose an arbitrary basis v = P i r i v ( I ) i ∈ L ( I ) ,where r i ∈ R are arbitrary real numbers, reshape v intoa matrix Γ , namely(Γ ) ab = v a ( N − b ;and then construct an hermitian matrix Γ = (Γ + Γ † ) + i (Γ − Γ † );(3) Diagonalize the class operators C of M ( H ), and theclass operators C ( s ) of its subgroup chain H ⊂ H ⊂ ... ⊂ H , and the matrix Γ simultaneously, CC ( s )Γ φ ( ν ) ε m = νmε φ ( ν ) ε m , then the eigenvectors φ ( ν ) ε m are the irreducible bases. Theeigenspace of ‘energy’ ε is an irreducible Rep space, thuswe can use the energy ε to label the multiplicity ( ν ) ε if theIPRep ( ν ) occurs more than once. The class operatorsare defined as the following[9, 21] C i = X h a ∈ H M ( h a ) M ( h i ) M † ( h a ) , (20)and C is a linear combination of C i with C = P i r i C i where r i ∈ R are arbitrary real numbers. The op-erators C ( s ) are defined in a similar way, which areused to lift the degeneracy of the eigenvalues and to re-duce the restricted Reps of the subgroups on the chain H ⊂ H ⊂ ... ⊂ H .In the first step, the eigenvectors of M ( h ) ⊗ M ∗ ( h ) , h ∈ H with eigenvalue 1 are required. When the dimension N of M ( h ) is large, it seems that one need to solve theeigenstates of matrices with dimension N . Actually, thiscomplexity can be avoided in two ways.One way is to obtain the eigenvectors of M ( h ) ⊗ M ∗ ( h )from the eigenstates of M ( h ) and M ∗ ( h ). Since the eigen-values of M ( h ) ⊗ M ∗ ( h ) are the product of the eigenval-ues of M ( h ) and M ∗ ( h ), the eigenvectors of the prod-uct matrix with eigenvalue 1 is the direct product of theeigenstates of M ( h ) and M ∗ ( h ) whose eigenvalues aremutually complex conjugate. For all the elements h ∈ H we can construct the eigenspace of M ( h ) ⊗ M ∗ ( h ) witheigenvalue 1 in the same way, then any state in the in-tersection of such eigenspaces satisfies the condition (1).The other way is to construct the matrix Γ directly,Γ = X h ∈ H M ( h ) AM † ( h ) , where A is an arbitrary square matrix[44]. Ob-viously above Γ satisfies the commutation relation M ( h )Γ M † ( h ) = Γ , which is equivalent to the eigenproblem M ( h ) ⊗ M ∗ ( h ) v = v with the vector v reshapedfrom Γ . Therefore, thus constructed matrix satisfiesthe conditions in step (1) and step (2). Practically thismethod is more straightforward. B. Reduction of Reps for anti-unitary groups
The same idea can be generalized to reduce generalReps of anti-unitary groups G = H + T H . For a gen-eral matrix Γ satisfying (4), (5) and (6), each of itseigenspaces is an irreducible projective Rep space of G .To lift the degeneracy of the eigenvalues of Γ , we canmake use of the class operators of H and those of itssubgroups.The central step is to construct the hermitian Hamilto-nian matrix Γ satisfying the restrictions (4), (5) and (6).We summarize the reduction procedure as the following:(1) Following the method in section III A, obtain a ma-trix Λ which is commuting with M ( h ) , h ∈ H , and thenconstruct a hermitian matrix Λ = (Λ + Λ † ) + i (Λ − Λ † );(2) Construct a matrix Γ from ΛΓ = Λ + M ( T ) K Λ KM † ( T )= Λ + M ( T )Λ ∗ M † ( T ) . It is easily verified that Γ M ( T ) K = M ( T ) K Γ because T ∈ H , M ( T )Λ = Λ M ( T ), and that [ M ( T ) K ] = ω ( T , T ) M ( T ). Furthermore, noticing that hT = T ( T − hT ) and that ( T − hT ) ∈ H , it can be shownthat Γ commutes with M ( h ) for all h ∈ H ;(3) Simultaneously diagonalize the class operators C of M ( H ) [see (20) for definition], the class operators C ( s )of the subgroup chain H ⊂ H ⊂ ... ⊂ H , and theHamiltonian matrix Γ, CC ( s )Γ φ ( ν ) ε m = νmε φ ( ν ) ε m , then the eigenvectors φ ( ν ) ε m are the irreducible bases,where the bases with the same ‘energy’ ε belong to thesame irreducible Rep-space ( ν ) ε .If IPReps with torsion number R = 4 are contained in M ( G ) after the reduction, then the restricted Rep of H ineach of the R = 4 IPRep is a direct sum of two identicalcopies of irreducible Reps of H . However, both Γ and C can only provide a single eigenvalue in the IRRep of G .Therefore the quantum number m in step (3) are doublydegenerate. In this case, we can use the hermitian matrixΛ to distinguish the two identical irreducible Reps of H .It is obvious that Λ commutes with Γ , C and C ( s ), so wecan add it to the commuting operators in step (3), C Λ , C ( s )Γ φ ( ν ) ε ε H ,m = νε H , mε φ ( ν ) ε ε H ,m , then all the degeneracies are lifted.Notice that we have used the class operators of H to define the class operator C for simplicity.The eigenvalues ν are not necessarily real (it is notreal if R = 2). In this case the eigenspaces of ν and ν ∗ belong to the same IPRep of G . One canalso adopt the class operators of the total group G , C i + = C h i + C T h i T − + C h − i + C T h i T − , C i − = i ( C h i + C T h i T − − C h − i − C T h i T − ) to construct C = P i ( r i + C i + + r i − C i − ) [21], where C h i is the classoperator of H in the restricted Rep and r i ± ∈ R arereal numbers. Then the eigenvalues of C are always realnumbers, but in this case the operators C ( s ) should in-clude the class operators of H and those of its subgroups. IV. APPLICATION OF THE HAMILTONIANAPPROACH IN PERTURBATION THEORY
The Hamiltonian approach can be generalized to ob-tain the response of the system to symmetry breakingprobe fields if the low-energy physics is dominated byparticle-like excitations, such as the electron-like quasi-particles in metals, Bogoliubov quasi-particles in super-conductors or the magnon excitations in the spin sector.We restrict our discussion to irreducible projective Repsof anti-unitary groups. A. k · p perturbation around high symmetry points In this section, we discuss the nodal-point and nodal-line structures in magnetic materials whose symmetrygroup are either type-III or type-IV Shubnikov magneticspace groups. The symmetry operations which keep amomentum kkk invariant (up to a reciprocal lattice vector)form a magnetic point group G ( kkk ) which is called thelittle co-group. The degeneracy of the energy bands at kkk is determined by the irreducible (projective) Reps of thelittle co-group. The dispersion around kkk can be obtainedusing the k · p perturbation theory.Suppose that the little co-group G ( kkk ) has a d -dimensional irreducible (projective) Rep, which is carried by the quasi-particle bases ψ αkkk , α = 1 , , ..., d , withˆ gψ † kkk ˆ g − = ψ † kkk M ( g ) K s ( g ) , (21)ˆ gψ kkk ˆ g − = K s ( g ) M ( g ) † ψ kkk , (22)for g ∈ G . The degeneracy is generally lifted at thevicinity of kkk . When δkkk is small enough, it is expectedthat ψ † kkk + δkkk and ψ kkk + δkkk vary in the way similar to (21)and (22) under the group action,ˆ gψ † kkk + δkkk ˆ g − = ψ † kkk +ˆ gδkkk M ( g ) K s ( g ) , (23)ˆ gψ kkk + δkkk ˆ g − = K s ( g ) M ( g ) † ψ kkk +ˆ gδkkk . (24)Suppose the Hamiltonian at kkk + δkkk is given by H kkk + δkkk = ψ † kkk + δkkk Γ( δkkk ) ψ kkk + δkkk , (25)where Γ( δkkk ) is an Hermitian matrix Γ † ( δkkk ) = Γ( δkkk ).When summing over all the momentum variation, thetotal Hamiltonian should preserve the G symmetry, i.e. ,ˆ g X δk H kkk + δkkk ! ˆ g − = X δk H kkk + δkkk ! , (26)for all g ∈ G . Substituting the equations (25), (23) and(24) into (26), we obtain, M ( g ) K s ( g ) Γ( g − δkkk ) K s ( g ) M † ( g ) = Γ( δkkk ) . (27)which is the most general symmetry requirement.If the leading order of Γ( δkkk ) is linear in δkkk , namely,Γ( δkkk ) ∼ δkkk · ΓΓΓ, then the dispersion around this high-degeneracy point froms a cone. For fermionic systems,a conic dispersion is called a Dirac cone[22–31] if d = 4and if ˜ T = I T ( I is the spacial inversion operation) is anelement of G ( kkk ) such that the energy bands are doublydegenerate away from kkk . On the other hand, if the de-generacy remains unchanged along a special line crossingthe point kkk , then this line is called a nodal line[32–42].Following the idea of the previous sections, here weprovide a criteria to judge whether the dispersion aroundthe point kkk is linear or of higher order, and whether thedegeneracy is stable in a high symmetry line.
1. Nodal points with linear dispersion
Firstly, we consider linear dispersion around kkk , namely,Γ( δkkk ) = X m =1 δk m Γ m + O ( δk ) . (28)Here δkkk is a dual vector under the point group opera-tions in H , namely,ˆ hδk m = X n D (¯ v ) mn ( h ) δk n , (29)where (¯ v ) is the dual Rep of the vector Rep ( v ) of theunitary subgroup H with D (¯ v ) ( h ) = (cid:0) [ D ( v ) ( h )] − (cid:1) T . Thevector Rep is real, so (¯ v ) is equivalent to ( v ) [in orthonor-mal bases, (¯ v ) is identical to ( v ), but we do not require thebases [ bbb , bbb , bbb ] in the reciprocal space to be orthonor-mal].From (27) ∼ (29), it can be shown (see Appendix C)that ΓΓΓ carries the dual vector Rep of H , namely, M ( h )Γ m M ( h ) † = X n D (¯ v ) nm ( h )Γ n . (30)In the following we first assume that the vector Rep ( v )is irreducible. The case ( v ) is reducible will be mentionedlater.According to the action of T on δkkk , we first discuss aspecial case where T acts trivially on δkkk , then go to thegeneral cases. The special case T δkkk = δkkk Firstly we consider the case that T acts trivially on δkkk , T δkkk = δkkk. (31)From (27), (28) and above equation, we have, M ( T ) K Γ m KM ( T ) † = Γ m . (32)The requirements (30) and (32) are similar to (4) and(5), respectively. If there exists three d × d Hermitianmatrices Γ , , satisfying these requirements, then thedispersion around kkk forms a cone. From the discussionin II B, we can judge the existence of Γ , , by checking ifthe projected space M ( h ) ⊗ M ∗ ( h ) P HT (or equivalentlythe projected space M ( h ) ⊗ F ( h ) P η ) contains the dualvector Rep (¯ v ) of H .When the vector Rep ( v ) of H is irreducible, then theexistence of linear dispersion can be checked by calculat-ing the following quantity, a H (¯ v ) = 12 | H | X h (cid:2) χ ( h ) χ ∗ ( h )+ ω ( hT , hT ) χ (( hT ) ) (cid:3) χ ( v ) ( h ) , (33) where [ χ (¯ v ) ( h )] ∗ = χ (¯ v ) ( h ) = χ ( v ) ( h ) has been used. If a H (¯ v ) is a nonzero integer, then the dispersion is linearalong all directions.The existence of Γ , , under the conditions (30) and(32) can also be checked straightforwardly by reducingthe product Rep M ( g ) ⊗ M ∗ ( g ) K s ( g ) , g ∈ G into directsum of IPReps using the method introduced in sectionIII B. If the resultant IPReps contain the dual vectorRep(s) whose bases are hermitian when reshaped intomatrix forms, then the leading order dispersion around kkk is linear. Therefore, we need a projection operator P H to project the bases carrying the dual vector Reps ontothe hermitian subspace (i.e. the union of real symmetricsubspace and the imaginary anti-symmetric subspace).Performing the projection P H and reshaping the remain-ing bases into hermitian matrices, then we obtain theexplicit form of Γ , , ,Γ m = p X i =1 r i γ mi , where p is the mutiplicity of the dual vector Rep(s) con-tained in the product Rep, r i ∈ R are arbitrary real num-bers, and γ , , i , i = 1 , ..., p are the bases of the i th dualvector Rep. Substituting into (28) and (25) we obtainthe k · p effective model.In next section, we will introduce an alternativemethod to obtain the hermitian matrices Γ , , withoutusing the projection operator P H . The general case
Generally, T acts on δkkk in the following way, T δk m = X n D (¯ v ) mn ( T ) δk n , where D (¯ v ) ( T ) is a 3 × D (¯ v ) ( T ) K can be considered as part of the dual vector Rep of theanti-unitary group G ( kkk ). Accordingly, Γ m should varyin the following way in anology to (32), M ( T )(Γ m ) ∗ M † ( T ) = X n Γ n D (¯ v ) nm ( T ) . (34)Similar to the discussion in Sec.II B 2, we introduce abases transformation ˜ ψ kkk = M ∗ ( T ) ψ kkk , thenˆ h ˜ ψ kkk ˆ h − = F T ( h ) ˜ ψ kkk , ˆ T ˜ ψ kkk ˆ T − = M T ( T ) K ˜ ψ kkk , where F ( h ) is defined in (9). We further define˜Γ m = Γ m M T ( T ) , (35)then (30) and (34) deform into M ( h )˜Γ m F T ( h ) = X n D (¯ v ) nm ( h )˜Γ n , (36) M ( T ) (cid:0) ˜Γ m (cid:1) ∗ M T ( T ) = X n D (¯ v ) nm ( T )˜Γ n , (37)respectively.Considering the set of matrices ˜Γ , , as a single col-umn vector ˜Γ with(˜Γ) n × d + i × d + j = (˜Γ n ) ij , (38)then ˜Γ carries the identity Rep of G , W ( h )˜Γ = ˜Γ , W ( T ) K ˜Γ = ˜Γ K, (39)where W ( h ) = D ( v ) ( h ) ⊗ V ( h ) = D ( v ) ( h ) ⊗ M ( h ) ⊗ F ( h ) ,W ( T ) = D ( v ) ( T ) ⊗ V ( T ) = D ( v ) ( T ) ⊗ M ( T ) ⊗ M ( T ) . Here the relation P m D (¯ v ) mi ( h ) D ( v ) mj ( h ) = δ ij has beenused. Later we will alternately use the notation ˜Γ m and˜Γ.Taking transpose of (35), we have(˜Γ m ) T = M ( T )(Γ m ) T = M ( T ) X n D ( v ) mn ( T ) M ∗ ( T )Γ n M T ( T )= η M ( σ ) X n D ( v ) mn ( T )˜Γ n , (40)where we have used the transpose of (34) namely P n (Γ n ) T D (¯ v ) nm ( T ) = M ∗ ( T )Γ m M T ( T ) and the hermi-tian condition Γ m = (Γ m ) † . Namely, the symmetry con-dition in (40) is a consequence of the requirements that˜Γ m should be hermitian and T -symmetric.Taking the transpose of ˜Γ twice, we obtain (cid:0) (˜Γ m ) T (cid:1) T = η P l D ( v ) ml ( σ ) M ( σ )˜Γ l M T ( σ ) = P l D ( v ) ml ( σ ) M ( σ )˜Γ l F T ( σ ). Therefore, the self-consistency requires that P l D ( v ) ml ( σ ) M ( σ )˜Γ l F T ( σ ) =˜Γ m , which is ensured by (36).Noticing ˜Γ m = [ η M ( σ ) P n D ( v ) mn ( T )˜Γ n ] T , we can de-fine the generalized twist operator which transforms ˜Γ m to [ η M ( σ ) P n D ( v ) mn ( T )˜Γ n ] T , T η = η ( I ⊗ T ) (cid:16) D ( v ) ( T ) ⊗ M ( σ ) ⊗ I (cid:17) = D ( v ) ( T ) ⊗ T η , where T η is defined in (18). Thus T η defines a general-ized ‘transpose’ of ˜Γ, and the self-consistency conditionis ( T η ) ˜Γ = W ( σ )˜Γ = ˜Γ. In the eigenspace of W ( σ )with eigenvalue 1, the operator T η has eigenvalues ± T η ˜Γ = ˜Γ (41)to be η -symmetric, then the projector onto the η -symmetric subspace is P η = 12 P σ ( I + T η ) , where P σ is the projector onto the eigenspace of W ( σ )with eigenvalue 1.Denoting P ( I ) as the projector onto the subspace ofidentity Reps P ( I ) = 1 | H | X h ∈ H W ( h ) , then obviously P ( I ) P σ = P ( I ) . The condition for the lin-ear dispersion is that the η -symmetric subspace containsthe identity Rep of H , namely a H ( I ) = Tr (cid:0) P ( I ) P η (cid:1) = 12 Tr (cid:0) P ( I ) ( I + T η ) (cid:1) ≥ . Defining the matrices W η ( h ) = 12 W ( h )( I + T η ) , with the matrix entries[ W η ( h )] mkl,nij = 12 h D ( v ) mn ( h ) M ki ( h ) F lj ( h )+ η X a D ( v ) mn ( hT ) M kj ( h ) F la ( h ) M ai ( σ ) i , then the condition of linear dispersion reduces to a H ( I ) =Tr ( P ( I ) P η ) = | H | P h ∈ H Tr W η ( h ) ≥
1. From theabove expression of W η ( h ), after some calculations (seeAppendix B) we obtain a H ( I ) = 12 | H | X h " | χ ( h ) | χ ( v ) ( h )+ χ ( v ) ( hT ) ω ( hT , hT ) χ (( hT ) ) . (42) Specially, if D ( v ) ( T ) = I , then χ ( v ) ( hT ) = χ ( v ) ( h ),above formula reduces to the equation (33); if D ( v ) ( T ) = − I , then χ ( v ) ( hT ) = − χ ( v ) ( h ), above formula can besimplified as a H (¯ v ) = 12 | H | X h (cid:2) χ ( h ) χ ∗ ( h ) − ω ( hT , hT ) χ (( hT ) ) (cid:3) χ ( v ) ( h ) . When the vector Rep ( v ) of G is reducible, then thedispersions may be different along different directions.In this case, we need to reduce the vector Rep ( v ) andcheck the resultant irreducible Reps one by one. Forinstance, if H = D h , then the vector Rep is reduced to( v ) = E u + A u , where ( k x , k y ) T vary in the rule of theRep E u and ( k z ) vary in the rule of A u . In this case,we need to replace ( v ) in (33) or (42) by ( E u ) and ( A u ).If a H ( E u ) is nonzero, then the dispersion along k x , k y islinear, otherwise the dispersion is quadratic or of higherorder. Similarly, if a H ( A u ) = 0, then the dispersion along k z is linear.
2. Procedure of obtaining the Γ , , matrices As a HI ≥
1, above procedure provides another way toobtain the matrices Γ , , besides the method of reducingthe product Rep M ( g ) ⊗ M ∗ ( g ) K s ( g ) , g ∈ G into directsum of IPReps. We only consider the case where the 3-dimensional vector Rep ( v ) is irreducible. The procedureis easily generalized to the cases in which ( v ) is reducible.0Firstly, obtain the eigenspace of P ( I ) P η = (cid:0) P ( I ) ( I + T η ) (cid:1) with eigenvalue 1. Supposing the dimension of thiseigenspace is p ≥
1, choose a set of orthonormal bases˜ ζ , ˜ ζ , ..., ˜ ζ p .Secondly, tune the bases in above subspace such that T is represented as IK . To this end, calculate the Repof T M ij ( T ) K = ˜ ζ † i W ( T ) K ˜ ζ j , and construct the new bases˜∆ j = X i ˜ ζ i M ij ( T ) . Then each of these new bases carries the identity Repsof the total group G (see Appendix D for details).Thirdly, from (38), we can decouple each eigenvector˜∆ i as ˜∆ i = 1 √ e ⊗ ˜ γ i + 1 √ e ⊗ ˜ γ i + 1 √ e ⊗ ˜ γ i , where e = (1 , , T , e = (0 , , T , e = (0 , , T stand for the bases aaa , aaa , aaa (not necessarily orthogo-nal) of the vector Rep ( v ) respectively, and ˜ γ mi is theSchmidt partner of e m . It can be shown that (see Ap-pendix D), ˜ γ mi satisfies the relations (36) and (37), andthat γ mi = ˜ γ mi M ∗ ( T ) is a hermitian matrices for any i = 1 , ...p and m = 1 , , i ,˜Γ = r ˜∆ + r ˜∆ + ... + r p ˜∆ p = 1 √ e ⊗ ˜Γ + 1 √ e ⊗ ˜Γ + 1 √ e ⊗ ˜Γ , (43)where r , ..., r p ∈ R are non-universal real constants.From (43) and the relation Γ m = ˜Γ m M ∗ ( T ), we obtainthe matrices Γ m ,Γ m = p X i =1 r i (cid:16) ˜ γ mi M ∗ ( T ) (cid:17) = p X i =1 r i γ mi .
3. Higher Order Dispersions and Nodal Lines
The discussion of linear dispersion can be straightfor-wardly generalized to higher order dispersions. Supposethat a set of order- N homogeneous polynomials P ( N ) i ( δkkk ) = X a + b + c = N f ( N ) i ( abc ) δk a δk b δk c carry a linear Rep (¯ µ ) of the group G , the existence ofthe dispersion H kkk + δkkk = X i P ( N ) i ( δkkk ) ψ † kkk + δkkk Γ ( N ) i ψ kkk + δkkk can be judged using the formula (42) with the vector Rep( v ) replaced by the linear Rep ( µ ) (see Appendix C foran example). The method of obtaining the correspondingmatrices Γ ( N ) i is also similar.If the vector Rep of G is reducible, it is possible thatthe degeneracy is lifted along some directions (such asthe k x , k y directions) but are preserved along certain di-rection (such as the k z direction) to form a nodal line.The little co-group on the line is generally smaller thanthe one on the conner of the BZ. If the IPRep of the littleco-group at the conner of the BZ is still irreducible alonga certain line, then this line is a nodal line. Therefore, theexistence of the nodal line can be judge from the formula(1) [42]. The same method can be applied to judge thestability of the degeneracy under external perturbations(see section IV B). B. Response to External Probe Fields
The IPRep M ( g ) K s ( g ) , g ∈ G of anti-unitary sym-metry group G results in energy degeneracy in single-particle spectrum. Here we discuss the possible liftingof the degeneracy under external probe fields, such as EEE and
BBB , stain, or temperature gradience, etc. We assumethat the probe fields carry irreducible linear Reps of thegroup G . For instance, electric fields EEE or magnetic fields
BBB carry vector Reps of the unitary subgroup H , but theyvary differently under the anti-unitary element T since EEE is invariant under time reversal while
BBB reverse its signunder time reversal.There are two possible consequences under externalprobes. The first possible result is that the degeneracyguaranteed by the IPRep M ( g ) K s ( g ) is preserved. Theother possibility is that the degeneracy is lifted in linearor higher order terms of the probe fields.To judge if the probe fields can lift the degeneracy ornot (summing over all orders of perturbation), we need toknow the remaining symmetry group with the presence ofthe perturbation, and then judge if the restrict Rep is re-ducible or not. Suppose the probe field reduces the sym-metry group from G = H + T H to G ′ = H ′ + T ′ H ′ where T ′ is anti-unitary. If the irreducible Rep M ( g ) K s ( g ) of G remains irreducible for G ′ , namely if1 | H ′ | X h ∈ H ′ (cid:2) χ ( h ) χ ∗ ( h ) + ω ( T ′ h, T ′ h ) χ (( T ′ h ) ) (cid:3) = 1 . holds for the group G ′ , then the degeneracy is robustagainst this perturbation.If the left hand side of above equation is not equal to 1,then the restricted Rep is reducible and the degeneracycan be lifted at certain order. In the following we onlydiscuss the linear splitting by external fields, such as EEE and
BBB . The linear response is given by the perturbed1Hamiltonian in form of H = X kkk ψ † kkk ( EEE · PPP + BBB · MMM ) ψ kkk (44)where P m , M m are CG matrices similar to the Γ m ma-trices discussed before. The existence of linear couplingterms PPP (or
MMM ) can be checked using the criterion (42)with M ( v ) ( T ) the transformation matrix of EEE (or
BBB )under the action of T . V. CONCLUSIONS AND DISCUSSIONS
In summary, from a physical approach, we derived thecondition (1) for the irreducible projective representa-tions of anti-unitary groups. This approach provides apractical method to reduce an arbitrary projective Repinto a direct sum of irreducible ones, which is applicablefor either unitary or anti-unitary groups.As a physical application of this approach, for singleparticle systems with magnetic space group symmetry,we provide the method to construct the k · p perturbationtheory at the high symmetry point of the Brillouin zone.We provide the criterion (42) to judge if the dispersionis linear or of higher order, and then provide the methodto obtain the corresponding k · p Hamiltonian up to a fewnon-universal constants.In the present work, we assume that the quasiparticlesvary under linear representations of the magnetic spacegroups. However, in strongly interacting systems, pro-jective representations of the magnetic space groups canemerge in the fractionalized low-energy quasiparticle ex-citations for systems with intrinsic topological order. Weleave the discussion of this situation for future study.
Acknowledgements
We thank L. J. Zou and Y. X.Zhao for helpful discussions. Z.Y.Y and Z.X.L. aresupported by the Ministry of Science and Technologyof China (Grant No. 2016YFA0300504), the NSF ofChina (Grants No.11574392 and No. 11974421), andthe Fundamental Research Funds for the Central Uni-versities and the Research Funds of Renmin Universityof China (Grant No. 19XNLG11). J. Yang and C.Fang are supported by Ministry of Science and Technol-ogy of China under grant number 2016YFA0302400, Na-tional Science Foundation of China under grant number11674370 and Chinese Academy of Sciences under grantnumber XXH13506-202 and XDB33000000.
Appendix A: Hermiticity of Γ and validity of (13)for general anti-unitary groups In this appendix, we firstly prove a lemma, then intro-duce a theorem. Type-I and type-II anti-unitary groupsare treated on equal footing.
Lemma 1
If the unitary representation M ( T ) K ofan anti-unitary element T satisfies the condition [ M ( T ) K ] = M ( T ) M ∗ ( T ) = I , then there ex-ist a unitary matrix U such that U † M ( T ) KU = U † M ( T ) U ∗ K = IK . Proof
Since [ M ( T )] † = [ M ( T )] − = [ M ( T )] ∗ , letting U = [ M ( T )] , we have U † = U ∗ = [ M ( T )] − . Conse-quently, one can easily verify that U † M ( T ) KU = U † M ( T ) U ∗ K = [ M ( T )] − M ( T )[ M ( T )] − K = IK.
Corollary 1
In a linear Rep of anti-unitary group G ,one can chose bases in the eigenspace of σ ≡ T witheigenvalue 1 such that T is represented as IK in thissubspace. Proof
Suppose D ( g ) K s ( g ) , g ∈ G is a N -dimensional lin-ear Rep of G , ∆ , ... ∆ N are the orthonormal bases of theeigenspace of σ = T with eigenvalue 1, namely. D ( σ )∆ i = ∆ i . Since σT = T σ , and accordingly D ( σ ) D ( T ) K = D ( T ) KD ( σ ) , we have D ( σ ) D ( T ) K ∆ i = D ( T ) K (cid:0) D ( σ )∆ i (cid:1) = D ( T ) K ∆ i , namely, the eigenspace of σ is closed under the action of T . In the eigenspace of σ with eigen value 1, the Rep of T takes the form M ( T ) ij K = ˜∆ † i D ( T ) K ˜∆ j , with M † ( T ) = M − ( T ) and M ( T ) M ∗ ( T ) = M ( σ ) = I . From lemma 1, the Rep of T can be trans-formed into IK in the new bases, ∆ ′ i = N X j =1 ∆ j M ji ( T ) , with M ( T ) K ∆ ′ i = M ( T )∆ ′ ∗ i K = ∆ ′ ∗ i K. Now it is ready to introduce the theorem.
Theorem 1 If ˜Γ is a common eigenvector ˜Γ of V η ( h ) , h ∈ H with eigenvalue 1, namely, V η ( h )˜Γ = ˜Γ for all h ∈ H , with V η ( h ) = [ V ( h )( I + T η )] and V ( h ) = M ( h ) ⊗ F ( h ) , then ˜Γ has the following prop-erties:1) it carries the identity Rep of H ;2) it is η -symmetric;3) if the basis satisfies V ( T )˜Γ ∗ = ˜Γ (i.e. if ˜Γ carries theidentity Rep of G ), then Γ = ˜Γ M ∗ ( T ) is an hermitianmatrix where ˜Γ has been reshaped into a matrix. Proof
Firstly, since V ( h ) = M ( h ) ⊗ F ( h ) is a linear Repof H , we define the following projection operator P ( I ) = 1 | H | X h ∈ H V ( h ) , which projects from the product space V ( h ) = M ( h ) ⊗ F ( h ) onto the identity Rep space. Accordingly, we have | H | X h ∈ H V η ( h ) = P ( I ) P η . Supposing ˜Γ is a common eigenvector of V η ( h ) , h ∈ H with eigenvalue 1, then we have | H | X h ∈ H V η ( h )˜Γ = P ( I ) P η ˜Γ = ˜Γ . (A1) Therefore, P ( I ) ˜Γ = P ( I ) P ( I ) P η ˜Γ = P ( I ) P η ˜Γ = ˜Γ , namely, ˜Γ is the CG coefficient coupling the direct prod-uct Rep V ( h ) = M ( h ) ⊗ F ( h ) to the identity Rep.On the other hand, from P ( I ) ˜Γ = ˜Γ and P ( I ) P η ˜Γ = P ( I ) ( I + T η )˜Γ = ˜Γ , we have
12 ( I + T η )˜Γ = ˜Γ . Therefore T η ˜Γ = ˜Γ . By definition, T η ˜Γ = [ η M ( σ )˜Γ] T ,so we have ˜Γ T = η M ( σ )˜Γ . Namely, ˜Γ is η -symmetric. Especially, for type-I anti-unitary groups, σ = E , the η -symmetry reduces to ˜Γ T = η ˜Γ .Thus we have verified that the eigenvector ˜Γ of P ( I ) P η ˜Γ = ˜Γ indeed has the properties 1) and 2).Now we illustrate that the relation V ( T ) K ˜Γ = ˜Γ canbe satisfied.Firstly we shows that V ( T ) K preserves the eigenspaceof P ( I ) . Noticing that T ( P h ∈ H h ) = ( P h ∈ H h ) T ,and that V ( g ) K s ( g ) , g ∈ G is a linear Rep of G , so V ( T ) KP ( I ) = P ( I ) V ( T ) K . Therefore, if ˜Γ is an eigen-vector of P ( I ) with P ( I ) ˜Γ = ˜Γ , then P ( I ) V ( T ) K ˜Γ = V ( T ) KP ( I ) ˜Γ = V ( T ) K ˜Γ . Namely, V ( T ) K ˜Γ is still an eigenvector of P ( I ) .Then we show that V ( T ) K also preserves theeigenspace of P η . From the definition of the unit twistoperator [( T ) kl,ij = δ kj δ li ] , it is easily to verify that ( X T ) kl,ij = X kl,ji and ( T X ) kl,ij = X lk,ij for arbitrary matrix X . Similarly, for a direct productmatrix ( X ⊗ Y ) kl,ij = X ki Y lj , the twist operator acts as ( [( X ⊗ Y ) T ] kl,ij = ( X ⊗ Y ) kl,ji = X kj Y li [ T ( X ⊗ Y )] kl,ij = ( X ⊗ Y ) lk,ij = X li Y kj , which gives [ T ( X ⊗ Y ) T ] kl,ij = [( X ⊗ Y ) T ] lk,ij = ( X ⊗ Y ) lk,ji = X lj Y ki = ( Y ⊗ X ) kl,ij . Since V ( T ) = M ( T ) ⊗ M ( T ) , we have [ T η V ( T ) K T η ]= η η ∗ T [ M ( σ ) ⊗ I ][ M ( T ) ⊗ M ( T )] T [ M ∗ ( σ ) ⊗ I ] K =[ I ⊗ M ( σ )][ M ( T ) ⊗ M ( T )][ M ∗ ( σ ) ⊗ I ] K =[ M ( T ) ⊗ M ( T )][ M ( σ ) ⊗ F ( σ )] ∗ K = V ( T ) K ( T η ) , (A2) where we have used ( T η ) = V ( σ ) . Therefore, if ˜Γ is η -symmetric, i.e. , if T η ˜Γ = ˜Γ , then T η V ( T ) K ˜Γ = T η V ( T ) K T η ˜Γ = V ( T ) K ( T η ) ˜Γ = V ( T ) K ˜Γ , whichmeans that (cid:0) V ( T ) K (cid:1) ˜Γ is still η -symmetric.Therefore, the eigenspace of P ( I ) P η with eigenvalue 1is preserved under the action of T . Namely, this eigspaceform a linear Rep of the anti-unitary group G . Noticingthat V ( σ ) = V ( T ) V ∗ ( T ) is represented as an identitymatrix in this eigenspace, from lemma 1, we can ‘diago-nalize’ V ( T ) K as IK in this subspace. Namely, we canchoose proper bases such that M ( T ) ⊗ M ( T )˜Γ ∗ = ˜Γ , orequivalently (11), holds.From the transpose of (11), we have M ( T )˜Γ † [ M ( T )] T = ˜Γ T = η M ( σ )˜Γ , where the symmetryequation has been used. Substituting Γ = ˜Γ M ∗ ( T ) intoabove equation and noticing M ( T ) M ∗ ( T ) = η M ( σ ) ,we finally obtain the hermitian condition Γ † = Γ . Thusthe property 3) has been verified. Noticing that the eigenspace of P ( I ) is either η -symmetric or ( − η )-symmetric, so P ( I ) P η is also a pro-jection operator (cid:0) P ( I ) P η (cid:1) = P ( I ) P η , therefore its eigenvalues are either 1 or 0. If M ( g ) K s ( g ) , g ∈ G is irreducible, then P ( I ) P η has onlyone nonzero eigenvalue. The trace of this projection op-erator Tr ( P ( I ) P η ) = 1 yields the irreduciblity condition(13). Appendix B: Derivation of (16) for Type-IIanti-unitary groups
Following the same discussion of type-I anti-unitarygroups, for type-II anti-unitary groups we obtain the ma-3trix form V η ( h ) = [ M ( h ) ⊗ F ( h )( I + T η )], namely[ V η ( h )] kl,ij = 12 (cid:16) [ M ( h ) ⊗ F ( h )] kl,ij + η [ M ( h ) ⊗ F ( h ) M ( σ )] kl,ji (cid:17) . (B1)From theorem 1 in appendix A, we can start with theequation (19), which can be expressed in terms of char-acters as1= 12 | H | X h,i,j (cid:16) M ii ( h ) F jj ( h )+ η M ij ( h )[ F ( h ) M ( σ )] ji (cid:17) . (B2)Remembering that M ( σ ) = η − M ( T ) M ∗ ( T ), the sec-ond term in (B2) can be transformed into X i,j η M ij ( h )[ F ( h ) M ( σ )] ji = η Tr[ M ( h ) F ( h ) M ( σ )]= η Tr[ M ( h ) F ( h ) η − M ( T ) M ∗ ( T )]=Tr[ M ( h ) M ( T ) M ∗ ( h ) M † ( T ) M ( T ) M ∗ ( T )]=Tr[ M ( h ) M ( T ) M ∗ ( h ) M ∗ ( T )]= ω ( h, T ) ω ∗ ( h, T )Tr[ M ( hT ) M ∗ ( hT )]= ω ( h, T ) ω ∗ ( h, T ) ω ( hT , hT ) χ (( hT ) ) . (B3)Finally, noticing Tr M ( h ) = χ ( h ) and Tr F ( h ) = χ ∗ ( h ),(B2) reduces to (16), namely,1 = 12 | H | X h ∈ H (cid:2) χ ( h ) χ ∗ ( h ) + ω ( hT , hT ) χ (( hT ) ) (cid:3) . (B4) Appendix C: k · p theory: Derivation of (30), (32),(34) and Discussion for General Dispersions We starts with the equation (27), namely, M ( g ) K s ( g ) Γ( g − δkkk ) K s ( g ) M † ( g ) = Γ( δkkk ) . Letting δkkk ′ = g − δkkk , then δkkk = gδkkk ′ and (27) becomes M ( g ) K s ( g ) Γ( δkkk ′ ) K s ( g ) M † ( g ) = Γ( gδkkk ′ ) . Since the summation over δkkk ′ is equivalent to the sum-mation over δkkk , therefore we have M ( g ) K s ( g ) Γ( δkkk ) K s ( g ) M † ( g ) = Γ( gδkkk ) . (C1)If there is a linear dispersion then Γ( δkkk ) = P m =1 δk m Γ m .Notice that δkkk varies as dual vector under the ac-tion of the unitary subgroup H , namely ˆ hδk m = P n D (¯ v ) mn ( h ) δk n . Substituting these relations into (C1)and letting g = h ∈ H , then we have M ( h ) X n Γ n δk n ! M † ( h ) = X m,n Γ m D (¯ v ) mn ( h ) δk n . (C2) Thus the equation (30) is proved, i.e. , M ( h )Γ n M ( h ) † = P m D (¯ v ) mn ( h )Γ m .Now consider the anti-unitary element g = T . From(C1), we obtain M ( T ) K Γ( δkkk ) KM † ( T ) = Γ( T δkkk ) . If T has a nontrivial action on δkkk , namely T δk m = P n D (¯ v ) mn ( T ) δk n , then linear dispersion Γ( δkkk ) = P m =1 δk m Γ m indicates that M ( T ) K X n Γ n δk n ! KM † ( T ) = X mn D (¯ v ) mn ( T )Γ m δk n , which is equivalent to (34), i.e. , M ( T )(Γ n ) ∗ M † ( T ) = P m Γ m D (¯ v ) mn ( T ) . Here we have used the fact that δk n ∈ R are real numbers. (32) is a special case of (34) with D (¯ v ) mn ( T ) = I .Similar discussion can be generalized to the case whenthe vector Rep is reducible, or to the cases where thedispersions are of higher order. Generally, the object P n Γ n δk n can be replaced by P i Γ ( N ) i P ( N ) i ( δkkk ), where P ( N ) i ( δkkk ) = X a + b + c = N f ( N ) i ( abc ) δk a δk b δk c , f abc ∈ R belongs to a set of order- N homogeneous polynomialsof δk , δk , δk which vary under the rule of irreduciblelinear Rep of G .For instance, in the case H = C v , the quadraticpolynomials ( P (2)1 , P (2)2 ) T = ( k x − k y , k x k y ) T vary as atwo-component column vector under the irreducible Rep( E ) = ( ¯ E ), namely, hP (2) i ( δkkk ) = X j D ( ¯ E ) ij ( h ) P (2) j ( δkkk ) , for h ∈ C v , and T P (2) i ( δkkk ) = X j D ( ¯ E ) ij ( T ) P (2) j ( δkkk ) . Accordingly, similar to (30) and (32) we have M ( h )Γ (2) i M ( h ) † = X j D ( ¯ E ) ji ( h )Γ (2) j ,M ( T )Γ (2) ∗ i M ( T ) † = X j D ( ¯ E ) ji ( T )Γ (2) j . The existence of quadratic dispersion terms with the form P i =1 Γ (2) i P (2) i ( δkkk ) can be judged using the formula (42)by replacing the vector Rep ( v ) with the linear Rep ( E ).Applying the method introduced in section IV A 1, wecan obtain the matrices Γ (2)1 , .4 Appendix D: k · p theory: hermiticity of Γ , , We define the projection operator P ( I ) = 1 | H | X h ∈ H W ( h )which project onto the subspace of identity Reps in theproduct Rep W ( h ) = D ( v ) ( h ) ⊗ M ( h ) ⊗ F ( h ). Similarly,1 | H | X h ∈ H W η ( h ) = P ( I ) P η . Following the discussion in Appendix A, the eigenvector˜Γ of P ( I ) P η with P ( I ) P η ˜Γ = ˜Γis also an eigenvector of P ( I ) and P η , namely, it is a η -symmetric vector which carries the identity Rep of H .Furthermore, referring to Appendix A and notic-ing the facts W ( T ) K P ( I ) = P ( I ) W ( T ) K and T η W ( T ) K T η = W ( T ) KW ( σ ) = W ( T ) K T η , itcan be verified that W ( T ) K preserves the eigenspace of P ( I ) and P η . Namely, the eigenspace L of P ( I ) P η witheigenvalue 1 is closed under the action of any g ∈ G ,hence forms a linear Rep space of G . From the lemma 1and its corollary, we can choose proper bases such thateach basis carries the identity Rep of G , namely, we canalways find the bases of L such that T is representedas M ( T ) K = IK .Now it is ready to prove that the matrices Γ , , constructed from ˜Γ are hermitian matrices. Corollary 2 If ˜Γ satisfy the relations (39) and (41),then the resultant matrices Γ , , are hermitian, where ˜Γ and Γ m are related by (˜Γ) n × d + i × d + j = (˜Γ n ) ij and Γ m = ˜Γ m M ∗ ( T ) . Proof
Equation (41) indicates that ˜Γ m satisfies the sym-metry condition (˜Γ m ) T = η X n D ( v ) mn ( T ) M ( σ )˜Γ n . (D1) Taking complex conjugation, above equation becomes (˜Γ m ) † = X n η ∗ D ( v ) ∗ mn ( T ) M ∗ ( σ )(˜Γ n ) ∗ = M ∗ ( T ) M ( T ) X n D ( v ) ∗ mn ( T )(˜Γ n ) ∗ , which yields M T ( T )(˜Γ m ) † = X n D ( v ) ∗ mn ( T ) M ( T )(˜Γ n ) ∗ . On the other hand, the second equation in (39) is equiv-alent to (37), which indicates that ˜Γ m M ∗ ( T ) = X k D ( v ) mk ( T ) M ( T )(˜Γ k ) ∗ . Since the vector Rep is a real Rep, comparing above twoequations we have M T ( T )(˜Γ m ) † = ˜Γ m M ∗ ( T ) , namely, (cid:0) Γ m (cid:1) † = Γ m . This completes the proof. [1] J. Schur, Journal f¨ur die reine und angewandteMathematik , 20 (01 Jan. 1904), URL .[2] F. Pollmann, A. M. Turner, E. Berg, and M. Os-hikawa, Phys. Rev. B , 064439 (2010), URL https://link.aps.org/doi/10.1103/PhysRevB.81.064439 .[3] X. Chen, Z.-C. Gu, and X.-G. Wen,Phys. Rev. B , 035107 (2011), URL https://link.aps.org/doi/10.1103/PhysRevB.83.035107 .[4] X. Chen, Z.-C. Gu, and X.-G. Wen,Phys. Rev. B , 235128 (2011), URL https://link.aps.org/doi/10.1103/PhysRevB.84.235128 .[5] X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G.Wen, Phys. Rev. B , 155114 (2013), URL https://link.aps.org/doi/10.1103/PhysRevB.87.155114 .[6] R.-J. Slager, A. Mesaros, V. Juriˇci´c, andJ. Zaanen, Nature Physics , 98 (2013), URL https://doi.org/10.1038/nphys2513 .[7] M. Barkeshli, P. Bonderson, M. Cheng, and Z. Wang,Physical Review B (2019), ISSN 2469-9969, URL http://dx.doi.org/10.1103/PhysRevB.100.115147 .[8] M. Hamermesh, Group theory and its appli-cation to physical problems , Dover Books onPhysics and Chemistry (Dover Publications,1989), ISBN 9780486661810,0486661814, URL http://gen.lib.rus.ec/book/index.php?md5=a86ba99acebcb82bc840426fab1b8efa .[9] J.-Q. Chen, M.-J. Gao, and G.-Q. Ma,Rev. Mod. Phys. , 211 (1985), URL https://link.aps.org/doi/10.1103/RevModPhys.57.211 .[10] M. Z. Hasan and C. L. Kane, Rev.Mod. Phys. , 3045 (2010), URL https://link.aps.org/doi/10.1103/RevModPhys.82.3045 .[11] X.-L. Qi and S.-C. Zhang, Rev.Mod. Phys. , 1057 (2011), URL https://link.aps.org/doi/10.1103/RevModPhys.83.1057 .[12] N. Read and D. Green, Phys.Rev. B , 10267 (2000), URL https://link.aps.org/doi/10.1103/PhysRevB.61.10267 .[13] X.-L. Qi, T. L. Hughes, S. Raghu, and S.-C.Zhang, Phys. Rev. Lett. , 187001 (2009), URL https://link.aps.org/doi/10.1103/PhysRevLett.102.187001 .[14] A. C. Christopher Bradley, The Mathematical Theoryof Symmetry in Solids: Representation Theory forPoint Groups and Space Groups , Oxford Classic Textsin the Physical Sciences (Oxford University Press,2010), 1st ed., ISBN 0199582580,9780199582587, URL http://gen.lib.rus.ec/book/index.php?md5=8aacfeeaa822a18b43c48f8a7973a304 .[15] R. Shaw and J. Lever, Communications inMathematical Physics , 257 (1974), URL https://doi.org/10.1007/BF01607948 .[16] S. K. Kim, Journal of Mathematical Physics ,197 (1984), https://doi.org/10.1063/1.526139, URL https://doi.org/10.1063/1.526139 .[17] J. Bardeen, The Journal of Chemical Physics ,367 (1938), https://doi.org/10.1063/1.1750270, URL https://doi.org/10.1063/1.1750270 .[18] S. F., Modern Theory of Solids(1987)(en)(736s) , INTERNATIONAL SE-RIES IN PHYSICS (McGraw-Hill, 1940), URL http://gen.lib.rus.ec/book/index.php?md5=8d029cfbf16e2e1f01e0d9c3f4ef80b8 .[19] I. Sakata, Journal of Mathematical Physics ,1702 (1974), https://doi.org/10.1063/1.1666528, URL https://doi.org/10.1063/1.1666528 .[20] R. Dirl, Journal of Mathematical Physics , 659(1979), https://doi.org/10.1063/1.524107, URL https://doi.org/10.1063/1.524107 .[21] J. Yang and Z.-X. Liu, Journal of Physics A: Math-ematical and Theoretical , 025207 (2017), URL https://doi.org/10.1088/1751-8121/aa971a .[22] S. M. Young, S. Zaheer, J. C. Y. Teo,C. L. Kane, E. J. Mele, and A. M. Rappe,Phys. Rev. Lett. , 140405 (2012), URL https://link.aps.org/doi/10.1103/PhysRevLett.108.140405 .[23] P. Tang, Q. Zhou, G. Xu, and S.-C. Zhang,Nature Physics , 1100 (2016), URL https://doi.org/10.1038/nphys3839 .[24] G. Hua, S. Nie, Z. Song, R. Yu, G. Xu, andK. Yao, Phys. Rev. B , 201116 (2018), URL https://link.aps.org/doi/10.1103/PhysRevB.98.201116 .[25] N. P. Armitage, E. J. Mele, and A. Vish-wanath, Rev. Mod. Phys. , 015001 (2018), URL https://link.aps.org/doi/10.1103/RevModPhys.90.015001 .[26] G. Hua, S. Nie, Z. Song, R. Yu, G. Xu, andK. Yao, Phys. Rev. B , 201116 (2018), URL https://link.aps.org/doi/10.1103/PhysRevB.98.201116 .[27] H. Watanabe, H. C. Po, and A. Vish-wanath, Science Advances (2018), URL https://advances.sciencemag.org/content/4/8/eaat8685 .[28] J. Cano, B. Bradlyn, and M. G. Vergniory, APL Materi-als , 101125 (2019), https://doi.org/10.1063/1.5124314,URL https://doi.org/10.1063/1.5124314 .[29] Y. Xu, L. Elcoro, Z.-D. Song, B. J. Wieder, M. G.Vergniory, N. Regnault, Y. Chen, C. Felser, andB. A. Bernevig, Nature , 702 (2020), URL https://doi.org/10.1038/s41586-020-2837-0 .[30] L. Elcoro, B. J. Wieder, Z. Song, Y. Xu, B. Bradlyn, andB. A. Bernevig, Magnetic topological quantum chemistry (2020), 2010.00598.[31] A. Bouhon, G. F. Lange, and R.-J. Slager,
Topologicalcorrespondence between magnetic space group represen-tations (2020), 2010.10536.[32] A. A. Burkov, M. D. Hook, and L. Ba-lents, Phys. Rev. B , 235126 (2011), URL https://link.aps.org/doi/10.1103/PhysRevB.84.235126 .[33] A. A. Burkov and L. Balents, Phys.Rev. Lett. , 127205 (2011), URL https://link.aps.org/doi/10.1103/PhysRevLett.107.127205 .[34] G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang,Phys. Rev. Lett. , 186806 (2011), URL https://link.aps.org/doi/10.1103/PhysRevLett.107.186806 .[35] Y. Chen, Y. Xie, S. A. Yang, H. Pan,F. Zhang, M. L. Cohen, and S. Zhang, NanoLetters , 6974 (2015), pMID: 26426355,https://doi.org/10.1021/acs.nanolett.5b02978, URL https://doi.org/10.1021/acs.nanolett.5b02978 .[36] C. Fang, Y. Chen, H.-Y. Kee, and L. Fu,Phys. Rev. B , 081201 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.92.081201 .[37] H. Weng, Y. Liang, Q. Xu, R. Yu, Z. Fang, X. Dai,and Y. Kawazoe, Phys. Rev. B , 045108 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.92.045108 .[38] T. Bzduˇsek, Q. Wu, A. R¨uegg, M. Sigrist, and A. A.Soluyanov, Nature , 75–78 (2016), ISSN 1476-4687,URL http://dx.doi.org/10.1038/nature19099 .[39] R. M. Geilhufe, F. Guinea, and V. Juriˇci´c,Phys. Rev. B , 020404 (2019), URL https://link.aps.org/doi/10.1103/PhysRevB.99.020404 .[40] D. Guo, P. Guo, S. Tan, M. Feng, L. Cao, Z. Liu, K. Liu,Z.-Y. Lu, and W. Ji, arXiv:2012.15218.[41] X. Cui, Y. Li, D. Guo, P. Guo, C. Lou, G. Mei, C. Lin,S. Tan, Z. Liu, K. Liu, et al., arXiv:2012.15220.[42] J. Yang, C. Fang, and Z.-X. Liu (2021),arXiv:2101.01733.[43] The linear combinations Γ ± = (Γ ± Γ † ) / †± = ± Γ ± , meaning that Γ + is Hermitian and Γ − is anti-Hermitian. So Γ − violates(6). On the other hand, if we transform Γ − into an her-mitian matrix i Γ − , then M ( T )( i Γ − ) ∗ M † ( T ) = − ( i Γ − ),namely, the hermitian matrix i Γ − forms an eigenstate of M ( T ) ⊗ M ∗ ( T ) K with eigenvalue −