Existence of bulk chiral fermions and crystal symmetry
EExistence of bulk chiral fermions and crystal symmetry
J. L. Ma˜nes
Departamento de F´ısica de la Materia CondensadaUniversidad del Pa´ıs Vasco, Apdo. 644, E-48080 Bilbao, Spain (Dated: April 2, 2012)We consider the existence of bulk chiral fermions around points of symmetry in the Brillouin zoneof nonmagnetic 3 D crystals with negligible spin-orbit interactions. We use group theory to showthat this is possible, but only for a reduced number of space groups and points of symmetry thatwe tabulate. Moreover, we show that for a handful of space groups the existence of bulk chiralfermions is not only possible but unavoidable, irrespective of the concrete crystal structure. Thusour tables can be used to look for bulk chiral fermions in a specific class of systems, namely thatof nonmagnetic 3 D crystals with sufficiently weak spin-orbit coupling. We also discuss the effectsof spin-orbit interactions and possible extensions of our approach to Weyl semimetals, crystals withmagnetic order, and systems with Dirac points with pseudospin 1 and 3 /
2. A simple tight-bindingmodel is used to illustrate some of the issues.
PACS numbers: 71.10.Ay, 71.15.Rf, 61.50.Ah, 37.10.Jk
I. INTRODUCTION
Electrons in the vicinity of the K points in graphene have linear dispersion relations and behave like mass-less chiral fermions. More concretely, the dynamics ofelectrons around these points is governed by the rel-ativistic, two-dimensional massless Dirac hamiltonian H ∼ σ x k x + σ y k y , and many of the exotic electronicproperties of graphene stem from this fact. This alsoturns graphene into a potential laboratory for two-dimensional relativistic dynamics, incorporating mass-less fermions, gauge fields and curved gravitational back-grounds. Moreover, optical lattices that could be usedto simulate relativistic systems with trapped cold atomscan be fashioned after graphene, with control over theproperties of the system. . Obviously three-dimensionalanalogs of graphene are potentially very interesting.Strictly speaking, the massless Dirac hamiltonian H describes only the low energy, orbital dynamics of elec-trons in graphene. As reviewed in Section III, spin-orbitcoupling in graphene gives fermions a very small mass. This mass is so small that for most practical purposesspin and orbital degrees of freedom decouple and H pro-vides an effective description of an enormous variety ofphenomena. In this paper we will consider 3 D analogs of graphene,i.e., crystals with orbital electron dynamics governedby the 3 D two-component massles Dirac hamiltonian H ∼ v(cid:126)σ · (cid:126)k , also known as the Weyl hamiltonian. Thishamiltonian describes massless chiral fermions, right-handed for v >
0, left-handed otherwise. Henceforth wewill use the term ‘orbital Weyl point’ to refer to pointsaround which the low energy dynamics in the absence ofspin-orbit couplings is described by the 3 D Weyl hamil-tonian, with an additional twofold degeneray due to elec-tron spin. We will also speak of ‘bulk chiral fermions’,keeping in mind that, as in graphene, they are exactlychiral in the limit of vanishing spin-orbit interactions.Also note that it is pseudospin , not electron spin, which is parallel (or antiparallel) to (cid:126)k in the chiral limit andthat, in these systems, pseudospin is purely orbital inorigin.These should be distinguished from other systemswhere the total hamiltonian, including electron spin andspin-orbit interactions, adopts the form of the Weylhamiltonian. These include surface states in topologi-cal insulators as well as novel three-dimensional ‘Weylsemimetals’. The spectrum of these systems, unlikegraphene and its 3 D analogs considered in this paper, re-mains gapless for arbitrary values of the spin-orbit cou-plings. Actually, some Weyl semimetals have strong spin-orbit interactions. Possible extensions of our methodsto Weyl semimetals will be considered in the last Section.It is well known that Weyl points have topologicalproperties, and no fine-tunning or symmetries areusually necessary for their existence. But symmetry,while not necessary, can sometimes be sufficient for theexistence of Weyl points. That is what we show in thispaper, where we investigate the role played by the spacegroups of crystals with time reversal symmetry (TRS) inthe existence of orbital Weyl points.The main results of this paper are summarized in Ta-bles I-II. Only crystals with one of the 19 space groupsin these tables can have orbital Weyl points at points ofsymmetry. Moreover, crystals with space groups in Ta-ble I must have orbital Weyl points at the listed points, irrespective of the actual crystal structure . For crystalswith space groups in Table II the situation is only slightlydifferent: At the listed points both orbital Weyl pointsand non-degenerate bands with quadratic dispersion re-lations are possible. These results are relevant to non-magnetic 3 D crystals with sufficiently weak spin-orbitinteractions and to cold atoms in optical lattices.The rest of the paper is organized as follows. Our mainresults, contained in Eq. (2) and Tables I-II are explainedin Section II. Section III considers the effects of spin-orbit interactions on the orbital Weyl points, and a sim-ple tight-binding model is constructed and analyzed in a r X i v : . [ c ond - m a t . s t r- e l ] M a r Section IV. Possible extensions of our approach to Weylsemimetals, crystals with magnetic order and other typesof Dirac points are considered in Section V. An outlineof the methods used to obtain Tables I-II is given in theAppendix.
II. ORBITAL WEYL POINTS AND CRYSTALSYMMETRY
Our strategy is based on the fact that the form of thehamiltonian in the vicinity of a point of symmetry (cid:126)K isstrongly constrained by the symmetries of the point inquestion. These include G K —the little group of the vector (cid:126)K — and combinations of TRS with spacegroup elements. Since we are interested in systems withvery weak spin-orbit interactions, we will consider firstthe structure of the orbital or spin-independent part ofthe hamiltonian. The transformation properties of or-bital wavefunctions are described by single-valued rep-resentations of the space group.As explained in the Appendix, we have carried out asurvey of all the single-valued irreducible representationsof the 230 space groups at points of symmetry in the Bril-louin zone (BZ). We find that, for most space groups,the constraints on the form of the hamiltonian aroundpoints of symmetry are incompatible with the structureof the Weyl hamiltonian. The comparatively few ex-ceptions are listed in Tables I-II. In all cases, two elec-tronic bands transforming according to a single-valuedirreducible representation (IR) of G K are degenerate atthe point of symmetry (cid:126)K . Near the point of symme-try, i.e., for (cid:126)K = (cid:126)K + (cid:126)k , the degeneracy is broken by (cid:126)k -dependent terms and the hamiltonian takes the form H ( (cid:126)k ) = v x σ x k x + v y σ y k y + v z σ z k z + O ( k ) (1)where v x = v y for uniaxial crystals and v x = v y = v z for cubic crystals. After appropriate rescalings of thecomponents of (cid:126)k for non-isotropic crystals, this is justthe Weyl hamiltonian H ∼ v(cid:126)σ · (cid:126)k , with the sign of v equal to the product of the signs of v i . The points ofsymmetry and IRs where this happens are listed in thelast column of Tables I-II in standard notation. TRS reverses the sign of (cid:126)K . In those cases where − (cid:126)K is not equivalent to (cid:126)K , we get a copy of the Weyl hamil-tonian at the mirror point − (cid:126)K and fermions have, inaddition to the pseudospin index associated to the Paulimatrices σ i , a ‘valley’ index. As shown in the Appendix,the total hamiltonian is then given by the 4 × H ( (cid:126)k ) = v (cid:18) (cid:126)σ · (cid:126)k (cid:126)σ · (cid:126)k (cid:19) + O ( k ) (2)This describes two degenerate massless fermions of the same chirality , right-handed for v >
0, left-handed oth-erwise. Somewhat surprisingly, we find that this doublingcontinues to take place even when − (cid:126)K ≡ (cid:126)K , i.e., for Space Group IRs I O P , P , P P O R , R , R P O R , R , R I T P , P , P P T R , R , R I D P P D A , A P D A , A I D W P D R TABLE I: Cubic, tetragonal and orthorhombic space groupswith orbital Weyl points. The small IRs are all 2 d (except for R of and , which is 4 d ) and refer to the symmetrypoints P ( , , ), R ( , , ), A ( , , ) and W ( , ¯14 , ¯14 ), withcomponents in the conventional basis of Ref. The stars havetwo vectors ( (cid:126)K , − (cid:126)K ) for body-centered ( I ) lattices, and asingle vector (cid:126)K ≡ − (cid:126)K for simple ( P ) lattices. TRS invariant momenta. In that case, Eq. (2) describestwo distinct fermions of the same chirality at the samepoint in the BZ. There is still a ‘valley’ index but, unlikein graphene, it can not be associated with two differentpoints in the BZ. This happens for the six space groupswith simple (P) Bravais lattices in Table I.The groups in Table I have one important feature incommon: The IRs in the last column of the table in-clude all the IRs at the point of symmetry. This meansthat, at that point, all the bands must form degener-ate pairs with Weyl hamiltonians. In other words, any crystal with space group in Table I will have bulk chiralfermions described by Eq. (2), irrespective of the concretecrystal structure. On the other hand, the space groupsin Table II have, besides the listed small IRs K and H ,which are two-dimensional and give rise to orbital Weylpoints, other one-dimensional small IRs ( K , K , H , H )not related to Weyl points. In this case, both orbitalWeyl points and non-degenerate bands with quadraticdispersion relations are possible at the listed points ofsymmetry. Space Group IRs P D K P D K , H P D K , H P D K P D K P D K , H P D K , H P D K , H P D K , H TABLE II: Hexagonal and trigonal space groups with orbitalWeyl points. The small IRs listed are all 2 d and refer to thesymmetry points K ( ¯13 , ,
0) and H ( ¯13 , , ), with componentsin the conventional basis of Ref. The stars have two vectors( (cid:126)K , − (cid:126)K ) in all cases. FIG. 1: Effects of spin-orbit coupling on the orbital Weylpoint of a cubic crystal (top) and graphene (bottom) in arbi-trary units.
We close this section by pointing out some special fea-tures in Tables I-II. The first one is that all the groupsin Table I are subgroups of the first entry, the cubicspace group I
32 and, despite the use of differ-ent conventional names (
P, R, A, W ), they also share thepoint of symmetry. Indeed, the cartesian coordinates forall the points in Table I can be written as (cid:126)K = ( πa , πb , πc ) (3)in terms of the unit cell constants, with b = c for uniaxialcrystals and a = b = c for cubic crystals. As a result,one can begin with any 3 D lattice with space group and reproduce all the cases in Table I by suitable defor-mations. The second somewhat surprising feature is thatall the stars have just one or two vectors. As a conse-quence, these crystals have only one or two orbital Weylpoints degenerate in energy. This should be contrasted,for instance, with the case studied in Ref., where 24Weyl points (away from points of symmetry) are presentat the Fermi energy. III. SPIN-ORBIT INTERACTIONS
Strictly speaking, our analysis so far applies only to‘spinless electrons’. It is well known that spin-orbit inter- actions in graphene open a gap and give fermionic excita-tions a small mass. In the case of graphene, the intrinsicspin-orbit hamiltonian is proportional to σ z ⊗ s z ⊗ τ z ,where s z and τ z are Pauli matrices for electron spinand valley indices respectively. Around each valley, thehamiltonian can be written in terms of 4 × H + H so = v ( α x k x + α y k y ) + β ∆ (4)where α i = σ i ⊗ s , β = ± σ z ⊗ s z and ∆ is the strength ofthe spin-orbit coupling. The matrices satisfy the Cliffordalgebra { α i , α j } = 2 δ ij , { α i , β } = 0 , β = 2 (5)This identifies Eq. (4) as the Dirac hamiltonian for4-component fermions with mass m = ∆ and spectrum E ± = ±√ ∆ + v k .For the space groups in Tables I-II, the most general k -independent spin-orbit hamiltonian compatible withspatial symmetries and TRS takes the form H so = 14 (∆ x σ x ⊗ s x +∆ y σ y ⊗ s y +∆ z σ z ⊗ s z ) ⊗ τ (6)where ∆ x = ∆ y for uniaxial crystals and ∆ x = ∆ y = ∆ z for cubic crystals. Actually, this valley independentform of the spin-orbit interaction is valid on the basis( e , e , ie ∗ , − ie ∗ ) of orbital wavefunctions introduced inthe Appendix. If one uses the more conventional basis( e , e , e ∗ , e ∗ ), then one has to append the valley matrix τ z to the x and z components in Eq. (6). Note that, inthe conventional basis, instead of Eq. (2), we would haveEq. (A6).The spectrum of the total hamiltonian v(cid:126)σ · (cid:126)k + H so canbe computed numerically and one finds that, in general,gaps are generated and all fermionic excitations acquiremasses. Cubic crystals, where H so is isotropic and de-pends on a single parameter ∆, are an exception and canbe treated analytically. In this case the spectrum is givenby E −± = − ∆4 ± (cid:114) ( ∆2 ) + v k E + ± = ∆4 ± vk (7)and contains massless excitations. This spectrum is rad-ically different from that of the Dirac hamiltonian ap-propriate for 2 D graphene. Note, in particular, that thelinear bands E + ± , together with E − + , form a triplet (seeFig. 1), following the usual rules for addition of angu-lar momenta with L = S = 1 / (cid:126)J = (cid:126)L + (cid:126)S . This isonly natural: assembling pseudospin and spin into a 4-component object is equivalent to taking the Kroneckerproduct of two j = 1 / SO (3) rotation group and this product decomposes ac-cording the rules of angular momentum composition. Thus, unlike in 2D where strong spin-orbit interactionssimply destroy the orbital Weyl points and turn masslessfermions into massive excitations, here we also get Diracpoints with J = 1. In the next Section we will presenta model that in the absence of spin-orbit has, besidesorbital Weyl points, Dirac points with pseudospin-one.The points with pseudospin-one would be split by strongspin-orbit interactions into Weyl points with J = 1 / J = 3 /
2. Subduction ofIRs can be used in principle to extend the analysisto non-cubic groups.Henceforth we will assume that spin-orbit interactionsare weak and can be ignored. In this limit the dynamics iswell described by the 3 D Weyl hamiltonian –albeit withan additional two-fold degeneracy due to spin– and, asa consequence, the systems considered in this paper mayshare some of the properties of Weyl semimetals.
IV. A TIGHT-BINDING EXAMPLE
As a practical application, we present a tight-bindingmodel with space group I O ), the first entry in Table I. According to our previous analysis, any latticewith this space group must have orbital Weyl points at P . Here we consider a lattice with four atoms per primi-tive unit cell, with cartesian coordinates (cid:126)r = a/ , , (cid:126)r = a/ , ¯1 , (cid:126)r = a/ , ¯1 ,
5) and (cid:126)r = a/ , , To each atom we associate a Bloch functionΦ i ( (cid:126)k ) = (cid:88) (cid:126)t(cid:15) T e i(cid:126)k · ( (cid:126)r i + (cid:126)t ) ϕ ( (cid:126)r − (cid:126)r i − (cid:126)t ) (8)where the sum runs over all the points in the Bravais lat-tice and ϕ ( (cid:126)r ) is an s -wave atomic orbital. Each atom hasthree nearest neighbors (NN), with bonds parallel to thethree cartesian planes. In terms of the reduced cartesiancomponents of the wave vector (cid:126)k = 2 π/a ( k x , k y , k z ) theNN tight-binding hamiltonian is given by H ( (cid:126)k ) = t e iπ ( k z − k y ) e iπ ( k y − k x ) e iπ ( k x − k z ) e − iπ ( k z − k y ) e iπ ( k x + k z ) e − iπ ( k x + k y ) e − iπ ( k y − k x ) e − iπ ( k x + k z ) e iπ ( k y + k z ) e − iπ ( k x − k z ) e iπ ( k x + k y ) e − iπ ( k y + k z ) (9) Γ P H N Γ +3-30 FIG. 2: Bands for the cubic lattice in the NN approximationfor t = −
1. The BCC Brillouin zone with its points and linesof symmetry can be seen in Fig. 3. where t < P is obvious inFig. 2. We can use standard group theory techniques to confirm that they actually are Weyl points, as pre-dicted. The four Bloch functions Φ i form the basis ofa reducible representation H that can be decomposedinto small IRs as H (Φ , . . . , Φ ) = P ( e , e )+ P ( u , u ).Up to normalizations, the symmetry-adapted vectors aregiven by e ∼ (1 , β , − , iβ ), e ∼ ( − β , − , − β , i ), The PostScript file with the Brillouin zone.
FIG. 3: Brillouin zone for BCC crystals. with β = (1 + √ − i ) /
2, and identical expressions for u , u with β replaced by β = (1 − √ − i ) /
2. Usinga unitary transformation U P to change to the symmetryadapted basis and expanding around the point P yields U † P H ( (cid:126)k ) U P = πt √ (cid:18) π + (cid:126)σ · (cid:126)k H ( (cid:126)k ) H † ( (cid:126)k ) − π − (cid:126)σ · (cid:126)k (cid:19) + O ( k )(10)where H ( (cid:126)k ) is given by H ( (cid:126)k ) = 1 √ (cid:18) k z ω ∗ k x − iωk y ω ∗ k x + iωk y − k z (cid:19) (11)with ω = e πi/ . There are thus two orbital Weyl pointsat P with different energies ± t √ which requires the net chirality of the BZ to vanish, al-though the fact that they appear at coincident points ispeculiar to this model. Note also the linear couplings be-tween the two points. Due to the split in energies, thesecouplings contribute corrections O ( k ) to the 2 × − (cid:126)K confirms that, due to TRS, each orbital Weyl pointis degenerate in energy with another point of the samechirality. Restoring the lattice constant yields a Fermivelocity v F = a | t | √ . Note however that, as no symmetryconnects the orbital Weyl points with different chirali-ties and energies, going beyond the NN approximation isexpected to give different Fermi velocities for them.Fig. 2 exhibits linear bands around the Γ and H pointsas well. Their nature is, however, very different from thatof the orbital Weyl points at P . Let’s consider, for thesake of concreteness, the Γ point. In this case, the rep-resentation associated with the Bloch functions decom-pose into IRs of dimension one and three, H = A + T .Transforming to the appropriate symmetry-adapted ba-sis and linearizing yields U † Γ H ( (cid:126)k ) U Γ = t − − iπk z iπk y iπk z − − iπk x − iπk y iπk x − + O ( k )(12)Up to a constant energy, the 3 × H T ( (cid:126)k ) = πt (cid:126)J · (cid:126)k (13)where ( J i ) jk = − iε ijk are spin-1 matrices. Thus, aroundthis Dirac point, electrons behave more like masslessspin- one particles, with spectrum E ( (cid:126)k ) = 0 , ± v F | (cid:126)k | and v F = a | t | . Indeed, one can check that, while the E = 0component is longitudinally polarized, the other two aretransverse, just like the propagating components of aphoton. Pseudospin-one Dirac points have been reportedin some two and three-dimensional systems.A look at Fig. 2 shows that, even if the Fermi levelcoincides with one of the orbital Weyl points at P , band Γ P H N Γ +3-30 FIG. 4: Bands for the cubic lattice with a tetragonal distor-tion ( (cid:15) = 0 .
4) for t = − .
25. See the main text for details. overlap will cause the dynamics to be dominated by largeelectron (or hole) pockets. The existence of band over-lap can be traced in this case to the 3-fold degeneraciesin Fig. 2, which force one of the bands arising from theorbital Weyl points to bend over. As 3 d IRs exist onlyfor cubic groups, we may modify the model by makingthe hopping parameters t ⊥ associated with bonds paral-lel to the OXY -plane different from the rest, t ⊥ = (cid:15) t .This reduces the symmetry to the tetragonal subgroup I
22 and eliminates the 3-fold degeneracies at Γ and H . Fig. 4 shows the bands for (cid:15) = 0 .
4. In the absenceof spin-orbit interactions the system would behave like agapless semiconductor with massless carriers for 1 / / V. DISCUSSION
In this paper we have studied the interplay be-tween crystal symmetry and the existence of bulk chi-ral fermions in nonmagnetic 3 D crystals with weak spin-orbit coupling. We have shown that the space groupplays a determinant role, as summarized in Tables I-II. AsWeyl semimetals depend on the existence of magnetic or-der or strong spin-orbit interactions, we have exploredan entirely different corner in the space of candidate 3 D crystals with bulk chiral fermions. We have also consid-ered the effects of spin-orbit interactions and shown that,unlike in 2 D , these may give rise to new critical pointssupporting massless fermions for different values of thepseudospin. For sufficiently weak spin-orbit interactionsthe dynamics is well described by the 3 D Weyl hamilto-nian and the systems considered in this paper may sharesome of the properties of Weyl semimetals.There are two obvious uses for the information in Ta-bles I-II. First, one can look for orbital Weyl points innonmagnetic crystals with weak spin-orbit interactionsand space groups in the tables, either in theoreticallycomputed electronic bands or experimentally. The otheruse is the design of 3 D lattices with Weyl points. Thismay allow for a physical realization of massless chiralfermions with cold atoms in optical lattices. Note thatthe physical relevance as well as the feasibility of detect-ing chiral fermions in actual crystals will be affected bystructure dependent features such as the position of theFermi level and the amount of band overlap. But know-ing that the bulk chiral fermions have to be there is ob-viously a good starting point. As shown in the exampleof Section IV, we can then try to engineer the requiredproperties by modifying the initial system.There are several possible extensions to this work. Oneis to consider the existence of orbital Weyl points awayfrom points of symmetry. All the points in a line of sym-metry, except for the endpoints, have the same group G K . Therefore, symmetry alone can not imply the exis-tence of bulk chiral fermions at a particular point in theline, although it will indicate whether this is possible atall. But it can, in some cases, imply the existence alongthe line of ‘semi-Dirac’ points, where the dispersion re-lations are linear only for some directions. For genericpoints in the BZ, symmetry alone has little to say and adifferent kind of analysis may be useful. Sometimes not two, but three bands become degener-ate at a Dirac point. This is the case with the Γ and H points in the example of Section IV. This is possi-ble for some cubic space groups, and group theory canbe used to determine the IRs and points of symmetrywhere this may happen. As shown in Section III, turn-ing on spin-orbit interactions will give rise to novel Diracpoints with pseudospin 3 /
2. Other, more complicate lin-ear hamiltonians are also possible and may be phys-ically relevant. Group theory can be used to classify oreven predict the existence of the different types of Diracpoints.In this paper we have analyzed all the single-valued ir-reducible representations at points of symmetry in theBZ. These are appropriate for orbital degrees of free-dom. By considering instead double-valued IRs wecould study the existence of Weyl points where the totalhamiltonian, including electron spin and spin-orbit in-teractions, takes the form of the 3 D Weyl hamiltonian.Thus, we could extend our analysis to TRS invariantWeyl semimetals. Double-valued IRs have been recentlyapplied to the study of ‘Dirac semimetals’. So far we have restricted ourselves to TRS invariantcrystals. The reasons are mostly practical. The sym-metries of crystals with magnetic order are classified bythe 1651 magnetic space groups , instead of the 230ordinary (Fedorov) space groups that classify crystalswith TRS. The amount of work required to examine thepoints of symmetry and irreducible corepresentations for all the magnetic groups is, obviously, much greater.Moreover, unlike ordinary space groups, there are fewdatabases with the magnetic space groups of crystals withmagnetic order.Nevertheless, some of the results in this paper can also be applied to spinless electrons in crystals with mag-netic order. The reason is that, by construction, the 19entries in Tables I-II describe situations where the 3 D Weyl hamiltonian is invariant under the ‘grey’ or ‘typeII’ Shubnikov space group associated to an ordinary(Fedorov) space group by the addition of the TRS oper-ation. Now, all the magnetic space groups derived fromthe Fedorov space group with the BNS settings aresubgroups of the grey group. As a consequence, the cor-responding Weyl hamiltonian is automatically invariantunder all the magnetic groups derived from the ordinaryspace groups listed in our tables. For instance, the Weylhamiltonians at the P point of group are automati-cally invariant under the derived magnetic groups . and . (in the BNS settings).One still has to check that the degeneracy of the twobands at the point of symmetry, necessary for the exis-tence of the Weyl point, is maintained under the lowersymmetry of the magnetic space group. If this is not thecase but the effects of the magnetic order are small, theWeyl point will survive, but move away from the point ofsymmetry. The main difference when dealing with crys-tals with magnetic order is that we can not exclude thepossibility of finding bulk chiral fermions around pointsof symmetry for space groups not listed in Tables I-II.That happens whenever the Weyl hamiltonian is invari-ant under the magnetic subgroup of the grey space group,but not under the grey space group itself. Acknowledgments
It is a pleasure to thank F. Guinea for useful com-ments on an earlier draft of this paper and to J. M.P´erez-Mato for help in using the Bilbao CrystallographicServer
Appendix A
In this Appendix we outline the methods used to ob-tain Tables I-II and Eq. (2). Symmetry operations be-longing to the space group G will be written g = { α | (cid:126)v } ,where α and (cid:126)v denote the rotation and translation partsrespectively. Let { e i ( (cid:126)K ) } be a basis of orbital wavefunctions that transform linearly under the action of G , e i ( (cid:126)K ) → e j ( α (cid:126)K ) R ji ( g ), where we sum over repeated in-dices. Invariance of the hamiltonian under G means that,for any wavefunction ψ , (cid:104) H (cid:105) ψ = (cid:104) H (cid:105) ψ g , where ψ g is thetransformed of ψ by the group element g . Expanding thiscondition on the basis { e i ( (cid:126)K ) } yields, in matrix notation R † ( g ) H ( α (cid:126)K ) R ( g ) = H ( (cid:126)K ) (A1)where H ij ( (cid:126)K ) = (cid:104) e i ( (cid:126)K ) | H | e j ( (cid:126)K ) (cid:105) . Time reversal θ is anantiunitary operation that acts on orbital wavefunctionsby complex conjugation, e i ( (cid:126)K ) → e i ( (cid:126)K ) ∗ = e j ( − (cid:126)K )Θ ji ,where Θ ij is a unitary matrix, and reverses the mo-mentum (cid:126)K . Invariance under θ impliesΘ † H ( − (cid:126)K )Θ = H ∗ ( (cid:126)K ) (A2)We will also have to consider combined antiunitary op-erations of the form θg . In this case invariance of thehamiltonian implies T † ( g ) H ( − α (cid:126)K ) T ( g ) = H ∗ ( (cid:126)K ) (A3)where T ( g ) = Θ R ∗ ( g ). Eqs. (A1,A2,A3) become pow-erful constraints on the form of the hamiltonian whenwe take (cid:126)K = (cid:126)K + (cid:126)k in the neighborhood of a point ofsymmetry (cid:126)K and consider a power expansion in (cid:126)k .Consider for instance Eq. (A1) with g restricted to thelittle group of (cid:126)K , G K , i.e., α (cid:126)K ≡ (cid:126)K R † ( g ) H ( (cid:126)K + α(cid:126)k ) R ( g ) = H ( (cid:126)K + (cid:126)k ) (A4)By definition, the matrix α belongs to the vector repre-sentation V . If the basis functions { e i ( (cid:126)K ) } belong tothe small IR R of G K , terms of order n in (cid:126)k in an expan-sion of the l.h.s. of Eq. (A4) will transform according tothe product R ∗ × R × [ V ] n , where R ∗ is the complex con-jugate of R and [ V ] n denotes the n -th symmetric powerof the representation V . Then one can use standardgroup theory techniques to determine, order by order in (cid:126)k , the most general form of the hamiltonian compatiblewith the symmetries of the vector (cid:126)K .In particular, a necessary condition for the existenceof the Weyl hamiltonian, which is linear in (cid:126)k , is thatthe vector representation V is contained in the product R ∗ × R for some 2 d IR R of G K . We have checkedthis condition on all the single-valued 2 d and 4 d IRs atthe points of symmetry of the 230 space groups to obtaina first list of candidates. Polar groups have been dis-carded from the outset, as they couple one component ofthe momentum to the unit matrix and the result is in-compatible with the structure of the Weyl hamiltonian.The amount of work involved at this stage has been sub-stantially reduced thanks to the use of ‘abstract groups’in Ref. In essence, the abstract groups represent classesof isomorphic little groups G K and their number is muchsmaller than the number of little groups.In the next step each candidate IR has been checked forinvariance of the corresponding hamiltonian under TRS. This involves using Eq. (A2) whenever − (cid:126)K ≡ (cid:126)K andEq. (A3) if (cid:126)K is not equivalent to − (cid:126)K but there existsa space group element g = { α | (cid:126)v } ∈ G such that − α (cid:126)K ≡ (cid:126)K . The IRs that pass this last test are listed in thelast column of Tables I-II. These IRs have an importantproperty: There is always a basis where the matrices R ( g )coincide, up to g -dependent phases, with the j = 1 / R / (ˆ nφ ) = exp (cid:18) − i(cid:126)σ · ˆ n φ (cid:19) (A5)where φ is the angle around the unit vector ˆ n . As a con-sequence, we can always transform to a basis where, upto appropriate rescalings of the components of (cid:126)k for non-isotropic crystals (see Eq. (1)), the linear hamiltoniantakes the standard Weyl form H ( (cid:126)K + (cid:126)k ) (cid:39) v (cid:126)σ · (cid:126)k .When − (cid:126)K is not equivalent to (cid:126)K , the basis at thesetwo points can be related by TRS. Choosing as basisat − (cid:126)K the complex conjugate of the one at (cid:126)K , i.e.,taking as 4 d basis ( e , e , e ∗ , e ∗ ), and using Eq. (A2)gives H ( − (cid:126)K + (cid:126)k ) (cid:39) − v (cid:126)σ ∗ · (cid:126)k . The off-diagonal blocksbetween (cid:126)K and − (cid:126)K vanish by translation invariance,and we have H ( (cid:126)k ) = v (cid:18) (cid:126)σ · (cid:126)k − (cid:126)σ ∗ · (cid:126)k (cid:19) + O ( k ) (A6)We can make the symmetry between (cid:126)K and − (cid:126)K ob-vious by using the SU (2) transformation σ y (cid:126)σ ∗ σ y = − (cid:126)σ to change the basis at − (cid:126)K so that the 4 d basis is( e , e , ie ∗ , − ie ∗ ). On this basis the hamiltonian takesthe form given in Eq. (2).The case − (cid:126)K ≡ (cid:126)K is more subtle, and we have twopossibilities. If R is a real 2 d IR, then TRS applies( e , e ) onto itself, with e ∗ i = e i . In this case, Θ = and Eq. (A2) requires H ( − (cid:126)k ) = H ( (cid:126)k ) ∗ , which is not sat-isfied by the Weyl hamiltonian. Thus 2 d real IRs areexcluded from Table I. For complex and pseudoreal 2 d IRs and for the real 4 d R of and , ( e ∗ , e ∗ ) arelinearly independent of ( e , e ). 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