Efficient Topological Materials Discovery Using Symmetry Indicators
EEfficient Topological Materials Discovery Using Symmetry Indicators
Feng Tang,
1, 2
Hoi Chun Po, Ashvin Vishwanath, and Xiangang Wan
1, 2 National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China Department of Physics, Harvard University, Cambridge, MA 02138, USA (Dated: May 21, 2018)Although the richness of spatial symmetries has led to a rapidly expanding inventory of possibletopological crystalline (TC) phases of electrons, physical realizations have been slow to materializedue to the practical difficulty to ascertaining band topology in realistic calculations. Here, weintegrate the recently established theory of symmetry indicators of band topology into first-principleband-structure calculations, and test it on a databases of previously synthesized crystals. Thecombined algorithm is found to efficiently unearth topological materials and predict topologicalproperties like protected surface states. On applying our algorithm to just 8 out of the 230 spacegroups, we already discover numerous materials candidates displaying a diversity of topologicalphenomena, which are simultaneously captured in a single sweep. The list includes recently proposedclasses of TC insulators that had no previous materials realization as well as other topologicalphases, including: (i) a screw-protected 3D TC insulator, β -MoTe , with gapped surfaces except for1D helical “hinge” states; (ii) a rotation-protected TC insulator BiBr with coexisting surface Diraccones and hinge states; (iii) non-centrosymmetric Z topological insulators undetectable using thewell-established parity criterion, AgXO (X=Na,K,Rb); (iv) a Dirac semimetal MgBi O ; (v) a Diracnodal-line semimetal AgF ; and (vi) a metal with three-fold degenerate band crossing near the Fermienergy, AuLiMgSn. Our work showcases how the recent theoretical insights on the fundamentals ofband structures can aid in the practical goal of discovering new topological materials. I. INTRODUCTION
Topological materials, exemplified by the topologicalinsulators (TIs) and nodal semimetals, showcase intrigu-ing physical properties which could not emerge had elec-trons been classical point-like particles [1–3]. Most ofthe known varieties of topological phenomena arising inweakly correlated materials require symmetry protection,and the inherent richness of spatial symmetries of crystalsmanifests as a corresponding diversity of distinct elec-tronic phases. Depending on the spatial symmetries atplay, an insulating 3D topological material could be char-acterized by surface states featuring protected conical orquadratic dispersions [4–7], interesting band connectivi-ties [8], or even 1D gapless modes pinned to the hingesof the crystal facets [9–15]. Moreover, spatial symmetriesalso stabilize band crossings and give rise to various kindsof nodal semimetals [3].There is much interest in finding concrete materialscandidates for the various topological crystalline (TC)phases. A conventional “designer” approach can be sum-marized as follows: first, one specifies a topological (crys-talline) phase of interest, and, typically through intu-ition, identifies possible materials classes which have theapproach symmetry and chemistry ingredients; second,one performs realistic first-principle electronic calcula-tions; third, one extracts the band topology from the cal-culation, either through the evaluation of wave-function-based invariants [1, 2] or, when applicable, through sim-ple criteria relating symmetry representations to specificforms of band topology [16–19]. Such criteria are exem-plified by the celebrated Fu-Kane parity criterion for Z TIs [16]. They offer immense computational advantage in the diagnosis of topological materials, as their evalua-tion only requires physical data defined on a small set ofisolated momenta.In the described conventional approach, the analysisstarts with a particular topological phase in mind. Cor-respondingly, a negative result only implies the targetedform of band topology is absent, but does not rule outthe existence of other forms of nontrivial band topology.With the ever-expanding catalogue of TC phases, suchoversight of band topology is nearly inevitable, and onemay have to re-analyze all the previously studied materi-als in the search of a newly proposed class of topological(crystalline) phase of matter.Here, we propose an alternative paradigm for the dis-covery of topological materials which can help overcom-ing the mentioned challenge. Instead of hunting for ma-terials candidates with a targeted form of band topologyin mind, our framework automatically singles out all thematerials featuring any form of robust band topology,provided that it is detectable using symmetry represen-tations . Our approach is inspired by the recent wave ofdevelopment on establishing a more comprehensive un-derstanding between symmetry representations and bandtopology [14, 15, 21–25]. In particular, our algorithm isbuilt upon the exhaustive theory of symmetry indica- By “robust” here, we refer in particular to band topology that isstable against the addition of trivial degrees of freedom, whichincludes all the usual topological phases with protected surfacestates. “Fragile topology,” as introduced in Ref. 20, is not robustin this sense, despite the fact that they can be detectable fromsymmetry representations. Such fragile phases will not be cap-tured in the materials search algorithm described in this work. a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y tors (SIs) developed in Ref. [22], which enables a reliablediscovery of topological materials even without a prioriknowledge on the precise form of the band topology in-volved. Remarkably, the same algorithm can be appliedto evaluate the SI of any materials in any space group,which leads to a versatile method well-suited for handlingthe diversity of TC phases.To demonstrate the power of this paradigm, we ap-ply our program and search for materials reported in adatabase of previously synthesized compounds [26], fo-cusing on 8 of the 230 space groups. We discover topo-logical materials featuring an array of distinct topologicalproperties, ranging from weak and strong topological in-sulators to the recently proposed “higher-order” topolog-ical crystalline insulators (TCIs) [9–15]. In addition, ouralgorithm also captures topological semimetals. In thefollowing, we will highlight four particular materials can-didates discovered: (i) β -MoTe , a screw-protected TCIwith helical 1D hinge states ; (ii) BiBr, a C -rotation pro-tected TCI [10] with 2D gapless Dirac-cone surface statescoexisting with 1D hinge states on the side; (iii) AgNaO,a non-centrosymmetric strong TI detected using a newSI utilizing the improper four-fold rotation S [14, 15];and (iv) Bi MgO , a Dirac semimetal with a clean Fermisurface consisting only of two symmetry-related Diracpoints. Other materials found, including weak TI, TCIwith hourglass surface states, semimetal with triply de-generate nodal point, and nodal-line semimetal, are de-scribed in the supplementary materials (SM) [27]. II. MATERIALS DIAGNOSIS THROUGHSYMMETRY INDICATORS
We will begin by briefly reviewing the theory of sym-metry indicators in Ref. 22, which is rested upon the ob-servation that, insofar as symmetry representations areconcerned, one can represent any set of bands of interestby a vector [18, 21]: n = ( ν, n k , n k , . . . n α k , . . . , n k , n k , . . . ,n α k , . . . , n k N , n k N , . . . , n α N k N , . . . ) , (1)where ν is the total number of the filled energy bands.The subscript k , k , . . . , k N runs over the distinctclasses of high-symmetry momenta in the Brillouin zone(BZ), and the superscript 1 , , . . . , α i , . . . labels the irre-ducible representation (irrep) of the little group ( G k i ) at k i . Here, n α i k i denotes the number of times an irrep α i of G k i appears within the bands of interest, and shouldbe integer-valued if the bands are separated from aboveand below by band gaps at all the momenta k i . However,if the Fermi energy intersects with the energy bands at Technically, these integer-valued “vectors” are not truly vectors;we will nonetheless abuse the terminology here to highlight thelinear structure they obey. a momentum k s , then the values of n α s k s are ill-defined.Nonetheless, one can still formally compute n α s k s usingstandard methods, and the valued obtained are generi-cally not integer-valued.While Eq. (1) is defined for a general set of bands,there are strong constraints on n if the bands correspondto a trivial atomic insulator (AI), i.e., one can construct afull set of symmetry-respecting Wannier orbitals, whosecenters fall into one of the Wyckoff positions and trans-form under symmetries according to the representationsof the corresponding site-symmetry groups. Conversely,any combination of Wyckoff positions with a choice onthe site-symmetry group representations gives rise to anAI. The data on Wyckoff positions and their associatedirreducible symmetry representations (irreps) have beenexhaustively tabulated [28–30]. This allows one to read-ily construct all the possible vectors n corresponding toan AI (see an example in the supplemental material (SM)[27]).As any AI is viewed as a vector, it can be expandedover a basis. More concretely, from the mentioned com-putation one can obtain d AI basis vectors { a i : i =1 , , . . . , d AI } , such that for any vector n AI arising froman AI, one can expand n AI = d AI X i =1 m i a i , (2)where m i are integers. The entries appearing in the ba-sis vectors a i are all integer-valued, and we denote thelargest common factor of the entries by C i . As, like anyvector space, the basis vectors are not uniquely defined,one may wonder if C i has any physical meaning. How-ever, there exist special choices of basis for which thevalues of C i are maximized, and such basis can be foundthrough the Smith normal decomposition. As concreteexamples, the AI basis vectors we use in this work, cho-sen to maximize C i , are all listed in the SM [27]. Un-like the basis vectors themselves, the set of maximizedcommon factors { C , . . . C d AI } is fixed for any symmetrysetting, and one may further label the basis vectors suchthat C ≤ C ≤ · · · ≤ C d AI .While our discussion so far is restricted to the sym-metry analysis of trivial atomic insulators, it forms theanchor for an efficient diagnosis of topological materialsachieved by integrating the described with first-principlecalculations. To see how, consider a material to bediagnosed. We first perform a routine band-structurecalculation to obtain n α k , the multiplicities of the sym-metry representations at the high-symmetry momenta k i for the filled bands. We then subject n to the sameexpansion of the AI basis a i as in Eq. (2) to arriveat a collection of expansion coefficients { q i } . If thematerials of interest are time-reversal invariant andfeatures significant spin-orbit coupling, the coefficients { q i } can be classified into the following three cases (thediscussion below has to be appropriately modified forother symmetry settings [24, 25]): TABLE I. We focus on the following space groups ( SG s), inwhich a strong topological insulators generate a Z subgroupin the group of symmetry indicators, X BS . The entry 2 ∈ Z corresponds to various kinds of topological crystalline insula-tors, and the predicted materials candidates for such phasesare tabulated. X BS Z × Z Z × Z Z × Z Z SG ,
12 166 61 , , Materials Ag F β -MoTe ,BiBr A7-P c-TiS Case 1
A band gap is found at all momenta k i and q i ’sare all integers.This indicates that n is formally indistinguishablefrom an AI as far as (stable) symmetry representationsare concerned. As such, the material at hand is eitheran atomic insulator, an accidental (semi-)metal, orpossesses band topology that is either indiscernible froma representation viewpoint [15, 17, 22] or is not stableagainst the addition of trivial degrees of freedom [20, 22].A more refined wave-function-based analysis is requiredif a more comprehensive understanding on the materialis desired. Case 2 : A band gap is found at all momenta k i but some q i ’s are not integers. Nonetheless, q i C i ’s are all integers.Barring the possibility of additional accidental bandcrossings, which one can readily check by computing theenergy (but not wave-functions) of the bands over theentire BZ, the material at hand is a band insulator withnontrivial band topology [14, 15, 22]. Furthermore, thetopological properties of the material can be exposedby evaluating the symmetry indicator (SI) r i ≡ q i C i mod C i , which takes value in 0 , , . . . , C i −
1. Note that,if C i = 1, then r i is necessarily 0 and conveys no in-formation. As we have arranged the integers C i in as-cending order, we can assume C i = 1 for all i < p and hence drop r , . . . , r p − from the discussion. TheSI can then be viewed as an element of the group [22] X BS ≡ Z C p × Z C p +1 × · · · × Z C d AI , and one can infer thepossible forms of band topology associated to the mate-rials from the results in Refs. 14 and 15. Note that, if X BS is trivial for a space group, then this case can neveroccur.We remark that, as we will see below, for some casesthere could be distinct forms of band topology givingrise to the same SI, and as such one has to furtherevaluate their associated band invariants to pinpoint itsconcrete topological properties. Case 3 : Not all of q i C i s are integers.As proven in Ref. 22, this indicates that the materialis a symmetry-protected (semi-)metal, where a (nodal)Fermi surface is unavoidable given the symmetries andfilling of the crystal. Note that this case can occur evenwhen X BS is trivial.From the discussion above, we see that cases 2 and 3 re-spectively correspond to promising materials candidates Space group symmetry analysislinear expansion
Material-specificrepresentationsGeneral atomic basis inspect
Case 1No topologydetected Case 2Topological(crystalline) insulator Case 3Topological(semi-)metals
Edgestates
FIG. 1. Topological materials discovery algorithm. The ex-pansion coefficients { q i } of the materials-specific representa-tion vector n against the atomic basis { a i } can be classifiedinto three cases. Cases 2 and 3 correspond respectively topromising materials candidates for topological (crystalline)insulators and semimetals. for topological crystalline insulators and (semi-)metals,with the only uncertainty being the detailed energeticswhich might bring about unwanted trivial bands near theFermi energy. Therefore, by screening materials throughcomputing their SIs, one can channel most of the com-putational effort to these promising candidates for an ef-ficient discovery of new topological material.In the following, we apply our search strategy (Fig. 1)to discover a variety of topological materials. In particu-lar, we will focus on experimentally synthesized crystals[26] with significant spin-orbit coupling and no magneticatoms. Furthermore, we restrict our attention to mate-rials with ≤
30 atoms in the primitive unit cell.We will focus our search to a number of space groupsfor which the SI group X BS is particularly interesting.First, we will focus on centrosymmetric space groups (i.e.,containing inversion), for which X BS (in the present sym-metry setting) contains a factor of Z . The generator1 ∈ Z corresponds to a strong TI, and the Z struc-ture, as discussed in Ref. 22, implies that taking twocopies of an inversion-symmetric strong TI results in anontrivial insulator even when all the conventional Z TI indices have been trivialized. It was later realizedthat such materials with SI 2 ∈ Z will always possesssymmetry-protected surface states [10, 14, 15], and theyshowcase a very rich phenomenology which depends onthe precise symmetry setting at hand: the sruface statescould be 2D in nature when protected by a mirror, glide,or a rotation symmetry, or may manifest as 1D hingestates when protected by inversion or screw symmetries(Fig. 2a). We will provide explicit materials candidatesfor TCIs with SI 2 ∈ Z , which are discovered throughour search among materials crystalizing in the centrosym-metric space groups ( SG s) , , , , , , . In particular, note that these space groups have X BS = ( Z ) j × Z with j = 0 , , , Z factors correspond to either weak TIs or weak mirrorChern insulators. In addition, we further consider thenon-centrosymmetric space group , which containsthe improper four-fold rotation S symmetry, which en-ables one to diagnose a strong TI through SI [14, 15, 22]even when the Fu-Kane criterion [16] is not applicable.Aside from TCIs corresponding to case 2 above, we alsoencountered numerous materials realizing case 3 above,which correspond to symmetry-protected (semi-)metals.We will also discuss the particularly promising candidatesfound. III. SCREW-PROTECTED HINGE-STATES IN β -MoTe Our first example is the transition-metal dichalco-genide (TMD) β -MoTe , discovered through a search ofmaterials crystalizing in space group ( SG ) ( P /m ).MoTe has three different structural phases: hexagonal α -phase [31], monoclinic β -phase [32] and orthorhom-bic γ -phase [33]. At room temperature, β -MoTe formsa monoclinic structure with SG [32]. Its monolayercrystal has been proposed as a candidate of quantumspin Hall insulator [34]. Furthermore, around 240–260 K it undergoes a structural transition to the non-centrosymmetric γ -phase [35]. The γ -phase was theoret-ically predicted to be a type-II Weyl semimetal and hasbeen verified experimentally [36–39]. Interestingly, wefound that β -MoTe in bulk crystal form is a 2 ∈ Z TCIin SG , which features 1D helical hinge surface statesprotected by a two-fold screw symmetry.As shown in Fig. 2b, in spite of the presence of elec-tron and hole pockets, β -MoTe has finite direct bandgap throughout the whole BZ. For SG , there are d AI = 5 AI basis vectors [22], which we denote by a i for i = 1 , . . . ,
5. Using the convention described in Sec.II, one finds C = C = 1, C = C = 2, and C = 4.This implies the SI group is X BS = Z × Z × Z (de-tails are shown in the SM [27]). Based on the electronicstructure calculation, we calculate the irrep multiplici-ties n α k s for the 56 valence bands, and obtain the rep-resentation vector n . Expanding n as in Eq. 2, we get( q , q , q , q , q ) = (12 , , , , ). Thus, β -MoTe has aSI of (0 , , ∈ Z × Z × Z , which indicates non-trivialband topology.As one can verify explicitly through the inversioneigenvalues of the bands, the SI (0 , ,
2) for SG im-plies the conventional Z TI invariants are all trivial[22]. Rather, the entry 2 ∈ Z can be interpretedthrough a recently introduced inversion topological in- ab c 𝑘 (cid:3053) 𝑐 = 𝜋 ab cd FIG. 2. (a) Representative topological (crystalline) phasescorresponding to the Z subgroup of the symmetry indicatorsconsidered. While materials with indicators 1 , ∈ Z corre-spond to conventional strong topological insulators, and that0 ∈ Z is consistent with a trivial phase, systems with indi-cator 2 ∈ Z can correspond to a diverse set of topologicalphases protected by crystalline symmetries. It is worth men-tioning that topological phase transition among these phasescan be obtained through parity switch (details are given inSM [27]). (b) The electronic structure of bulk β -MoTe fromfirst principles calculation. (c) Geometric setting for the cal-culation in (d), where open boundary conditions along thea- and b-axes with lengths of 20 and 50 lattice constants re-spectively, and periodic boundary condition is imposed alongthe screw axis (i.e., c -axis). (d) The electronic structure onthe prism depicted in (c). The red lines correspond to thedispersions of the hinge states (each is two-fold degenerate).Inset shows the same plot but for a different energy window,showing in particular the expected Kramers degeneracy at k z = π/c . We find there are four bands whose eigenstatesare mainly localized at two of the four hinges between the(100) and (010) planes, as indicated by thick red lines in (c).These hinge states showcase a characteristic spin-momentumlocking indicating their helical nature, as expected from thetheoretical predictions [14, 15]. This is shown schematicallyby the colored arrows in (c), which denote to the direction ofmotion for the opposite spins. variant κ [14, 15]. The SI alone, however, does notuniquely determine the precise form of band topology for SG [14, 15], since it can correspond to either a mirror-protected TCI characterized by a nontrivial mirror Chernnumber (MCN), or a 2 screw-protected TCI with char-acteristic hinge-surface states. These two scenarios canbe further distinguished by computing the MCN on thetwo planes with k z = 0 and k z = π/c . Our results, com-puted through the WIEN2k package [40], show that theMCNs vanish for both planes (see SM for details [27]).Therefore, we conclude β -MoTe is a candidate for the2 screw-protected TCI.To verify the above theoretical predictions, we studythe electronic structure on an inversion symmetric prismas shown in Fig. 2c. We impose open boundary condi-tions along two directions, and periodic boundary con-dition along the screw axis. Due to the large size of thesupercell, we compute the band structure through a tightbinding (TB) model by considering orthogonal Mo’s d or-bitals and Te’s p orbitals (details are shown in the SM[27]). Not only owning the same topological feature (i.e.the same n α k s , κ and MCNs), the TB model also reason-ably reproduces the bulk electronic structure of β -MoTe [27]. The electronic structure on the prism is shown inFig. 2d. From the spatial profile of the eigenstates, weidentity four modes mainly localized at two of the hingesbetween (100) and (010) planes (Fig. 2c), which corre-spond to the helical 1D hinge states. IV. C -PROTECTED DIRAC SURFACE STATESAND COEXISTING HINGE STATES IN BiBr Next, we consider the monoclinic SG ( C /m ), whichcontains a (symmorphic) C rotation symmetry. Unlikethe 2 screw, which is not respected on any surface, C isrespected on the surface normal to the rotation axis, andhence one expects 2D gapless surface states on for C -protected topological phases. Furthermore, it was foundthat such 2D surface states coexist with 1D hinge states[10].Similar to the previous discussion, we first compute theAI basis vectors, from which we recover d AI = 7 and theSI group X SG = Z × Z × Z [22, 27]. This is thenused in the materials search, and we identify BiBr [41] asa TCI with a SI of (0 , , ∼ .
24 eV), as shownin Fig. 3a [27]. This SI guarantees that the conventional Z TI indices all vanish [22], and it remains to determineif the band topology is protected by mirror or the C rotation symmetry [14, 15]. To this end, we computedthe MCNs for the mirror planes k z = 0 and k z = πc , andfound that they both vanish. This establishes BiBr as afirst example of the C -rotation protected TCI.To further analyze the properties of BiBr, we constructa TB model starting with the p orbitals for both Biand Br [27]. Reproducing the electronic structures well,our TB model is also topologically equivalent to thatof the first principles calculation results. To verify thepresence of hinge states, we take an inversion-symmetricprism geometry similar to Fig. 2c, with periodic bound-ary condition along the (0 , ,
1) direction (i.e. k z is con-served) but open boundary conditions for the (0 , ,
0) and(1 / , , /
2) directions (the directions are denoted usingthe conventional lattice basis vectors [27]). The bandstructure of the TB model on the prism is shown in Fig.3(b). As expected, a pair of helical hinge states are found.In addition, two symmetry-related Dirac cones pre-dicted on the (0 , ,
1) surface of this TCI [10]. Thisis verified explicitly by computing the electronic struc-ture in a slab geometry. The results are shown in Fig. meV ab cd
FIG. 3. (a) Electronic structure of bulk BiBr from densityfunctional calculation. (b) The hinge states along c directionof a centrosymmetric prism for BiBr; the red lines denotehinge states traversing the bulk gap. Similar to the β -MoTe case, they are mainly localized at opposite hinges related by C rotation. (c) (001)-surface states for BiBr based on atight-binding model. Two symmetry-protected Dirac conesare found, as indicated by the black arrows. (d) A line cut of(b) at k = 0 . k = − . C -protected surface Dirac cones. On each surface, there aretwo Dirac cones located at the generic, but C -related,pair of points (( k D , k D ) = ± ( − . , . V. NON-CENTROSYMMETRIC STRONG TIAgNaO
While the Fu-Kane parity criterion [16] is instrumentalin the discovery of Z TIs, it is not applicable to crystalswithout inversion symmetry. Since a Weyl semimetallicphase is expected to intervene a trivial-topological phasetransition in the absence of inversion symmetry [42], it isalso of great interest to discover TI materials classes innon-centrosymmetric SG s. Curiously, unlike the conven-tional approach where the inapplicability of the Fu-Kanecriterion demands a wave-function-based computation ofthe Z invariants [43, 44], it was found that the improperfour-fold rotation S allows one to diagnose the strongTI index using SI [14, 15].In view of this, we apply our search algorithm tothe SG ( F ¯43 m ), which are non-centrosymmetricand contains S in the point group. We find AgXO(X=Na,K,Rb) [45] to be insulators with the SI 1 ∈ Z .In the following, we take AgNaO as an example, withthe corresponding discussions for AgKO and AgRbO rel-egated to the SM [27]. As shown in Fig. 4, AgNaO hasa full gap ( ∼ meV ). Its strong TI nature can also beunderstood from its orbital characters: throughout theBZ, the s ( d ) band is mainly above (below) the Fermilevel. However, there is a band inversion near the Γ point,where the s -like Γ bands are below the d -like Γ bands(Fig. 4) . As shown in the SM [27], this band-inversionpattern results in an S invariant, as defined in Refs. 14and 15, of κ = 1, which is consistent with the claim thatit is a strong TI. FIG. 4. Electronic structure of the non-centrosymmetricstrong topological insulator AgNaO. Γ , and Γ label thetwo 2D irreps and one 4D irrep, respectively. VI. DIRAC SEMIMETAL MgBi O As illustrated in Sec. II and highlighted in Fig. 1, ouralgorithm is not limited to discovery topological (crys-talline) insulators, but is also well-suited to unearthingtopological semimetals. Here, we report our discoverythat MgBi O is a Dirac semimetal with a nodal Fermisurface comprising only of a pair of symmetry-relatedDirac points.MgBi O crystalizes in the centrosymmetric SG ( P /mnm ) [46]. We found that its expansion coeffi-cients { q i } fall into case 3 of Fig. 1, which dictates nec-essarily gaplessness at the Fermi surface. Further bandstructure calculation reveals that the symmetry-dictatedband crossings locate exactly at the Fermi energy with-out any accompanying trivial bands. The stability ofthe band crossing points can be understood as follows:near the Fermi level, the low energy electronic behav-ior is mainly determined by Bi-6 s states (conductionbands) and O-2 p states (valence bands). Along the high-symmetry line Γ-Z, the Bi-6 s and O-2 p respectively fur-nish the Λ and Λ representations of the little group C v .In addition, there is a band inversion ( ∼ . k · p analysis that the band crossing leads to a Diracpoint. DP FIG. 5. Electronic structure of Dirac semimetal MgBi O .Along the Γ- Z line, there is a Dirac point originated from thecrossing of two different 2D irreps (Λ and Λ , indicated byblack and blue lines respectively) which is protected by C v symmetry. VII. OTHER TOPOLOGICAL MATERIALSDISCOVERED
Other ∈ Z materials. Besides β -MoTe and BiBr,we also find three other materials candidates with a SIof 2 ∈ Z ≤ X BS . We will summarize the findings below,and relegate the detailed analysis to the SM [27].First, we found that the cubic crystal TiS [47]( SG ) is a glide-protected TCI with hourglass surfacestates. Second, elemental phosphorus in the A7 structure( SG ), which occurs at about 9GPa [48], is predictedto be an inversion-protected TCI with 1D hinge states,akin to the recently realized case of elemental bismuth[49]. Finally, we find that Ag F [50] ( SG ) is a weak TIwith additional inversion-protected band topology char-acterized by the invariant κ = 2. Centrosymmetric strong TIs.
Our search algorithmalso naturally identifies conventional strong TIs diagnos-able through the Fu-Kane parity criterion [16]. We iden-tify CaAs ( SG ), Bi PbTe ( SG ) and CaGa As ( SG ) are all STIs [27]. Other types of topological semimetals.
We also foundother topological semimetals corresponding to case 3 de-scribed in Sec. II. In particular, AuLiMgSn and AgF are particularly interesting, with the former featuringsymmetry-enforced band crossings leading to three-folddegenerate gaplessness points, albeit at ∼ . VIII. CONCLUSION
We proposed an efficient and versatile method tosearch for topological (crystalline) materials by combin-ing the theory of symmetry indicators of band topologycombined with first principles calculations. Focusingon merely 8 space groups, we discovered numeroustopological materials featuring a diverse set of bandtopology, which ranges from conventional Z topologicalinsulators and nodal semimetals to recently proposedphases of topological crystalline insulators. Many moresymmetry settings are yet to be explored. To wit, thereare 230 space groups, and even 1 ,
651 magnetic spacegroups, for which the symmetry indicators have beencompletely derived [22, 24]. It is of great interest toapply our materials search strategy to these other sym-metry settings, where many more topological materialslikely reside.
Note Added:
While completing this manuscript, Ref. 51appeared, which proposed a candidate material for arotation-symmetry protected TCI confirmed using sym- metry indicators.
ACKNOWLEDGMENTS
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CONTENTS
I. Detailed descriptions on implementation of SI method. 1II. The detailed discussions on materials highlighted in the main text 2A. β -MoTe k · p model for the Dirac semimetal in the main text 7III. The other “2 in Z ” materials 9A. Hourglass insulator TiS F , Z C M in β -MoTe and BiBr 16VII. Tight binding model for MoTe and BiBr 17VIII. Method 17IX. Calculation of atomic insulator basis 17X. Reduction of AI on HSPs 19XI. Atomic basis vectors 20References 20 I. DETAILED DESCRIPTIONS ON IMPLEMENTATION OF SI METHOD.
Symmetry indicators (SIs) [1] of band topology are very powerful for the efficient diagnosis and prediction oftopological materials based on the first principles calculations. In this section, we will give detailed descriptions onhow to implement the SI method to a real material, and diagnose it by first principles calculations. Other than anindicator of band topology for the insulator, SI method is also very powerful to diagnose the topological semimetal(e.g. Dirac semimetal (DSM), multiple-fold degenerate semimetal, nodal-line semimetal, etc.) as we will show in thefollowing section.One first performs a routine first principles electronic structure calculation for a material. Then according to thespace group ( SG ), one can obtain the irreducible representations (irreps) of its valence bands (i.e. the first ν bands,where ν is the number of valence electrons in the primitive unit cell) at all the high symmetry points (HSPs) in theBrillouin zone (BZ). We denote the little group at the i th HSP, namely k i point, as G ( k i ). The α i th ( α i = 1 , , . . . )irreducible representation (irrep) of G ( k i ) is labeled by α i . If the valence bands and conduction bands are separatedthroughout the BZ , the electronic structure can be described by the number of the occurrences for the α i th irrep a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y in the valence electronic bands, i.e. n α i k i . The symmetry content of such valence bands is dubbed a “band structure”(BS) in Ref. 1. The BS can be represented by an integer-valued vector, n =:( ν, n k , n k , . . . , n α k , . . . , n r k , n k , n k , . . . , n α k . . . , n r k , (1) n k i , . . . , n α i k i , . . . , n r i k i , · · · , n k N , . . . , n α N k N , . . . , n r N k N ) . (2)In the above equation, k i denotes the HSP as before, where i takes 1 , , . . . , N ( N is the total number of HSPs). Thesuperscript labels the irrep for the corresponding G ( k i ), α i = 1 , , . . . , r i and the number of these irreps is r i for the i th HSP. All the HSPs and their irreps for the 230 SG s can be found in Ref. [2].Suppose that the valence and conduction bands touch at the i th HSP (i.e. k i ), and these touching bands formthe j th irrep of G ( k i ). We can still obtain n j k i by the standard method. In this case n j k i is not generally an integer.This belongs to Case 3 in the main text. Furthermore, even if all the n α k ’s are integers, they may not satisfy all thecompatibility relations, i.e., there will be some symmetry-enforced band crossing(s) in high symmetry line or plane.This also belongs to Case 3 of the main text. When there is an indicator of the band crossing, we then need tocarefully analyze the position(s) of the band crossing(s).On the other side, it is clear that the atomic insulator (AI), namely a group of electronic bands adiabaticallyconnected to a strict atomic limit, forms a BS. Ref. 1 has shown that there are d SG AI AI basis vectors (i.e. a SG i )for any given SG , i.e. any AI in this SG can be expressed by a linear combination of d SG AI AI basis vectors [1]: n AI = P d SG AI i =1 m i a SG i , where m i s are all integers [1] as Eq. 2 of the main text. (In Sec. IX , we give the detaileddescriptions on how to calculate the AI basis vectors for a given SG . We also list the AI basis vectors for all the SG s encountered in the work in Sec. XI).In addition to AI, Ref. 1 also found that any BS in SG can also be expanded on the AI basis vectors: n = d SG AI X i =1 q i a SG i , (3)however here the expansion coefficients q i s can be non-integers as we show in the main text. This is because thatsome AI basis vector a i may have a common factor C i for all its entries, so q i can be a rational number only requiringthat q i C i is an integer. When all the q i s are integers, the BS is indistinguishable from an AI as far as symmetryrepresentations are concerned (in the “stable sense” as elaborated in Ref. 3). It should be emphasized that this alonedoes not preclude the existence of band topology; rather, it simply implies more refined methods are required todetect, or rule out, band topology in the system.In contrast, when some q i ’s are not integers but q i C i ’s are all integers, up to detailed energetics the system is atopological (crystalline) insulator (here, as in the main text, we consider systems with time-reversal symmetry andstrong spin-orbit coupling; the statements have to be modified accordingly in other symmetry settings [4, 5]).The AI basis vectors for any SG can be easily obtained, and the first principles calculations for n α k is computationallyeasy as it only involves wave-function data at a small number of isolated momenta. Thus, simply by analyzing theexpansion coefficients q i s , it is highly efficient to screen crystal-structure databases and diagnose the topological(crystalline) insulators or (semi-)metals, following the flow described in Fig. 1 in the main text. II. THE DETAILED DISCUSSIONS ON MATERIALS HIGHLIGHTED IN THE MAIN TEXTA. β -MoTe The β -MoTe [6] crystallizes in the primitive monoclinic Bravais lattice. Its SG is No. 11. The crystal structureis shown in Fig. 1(a). We adopt such a setting that c is the unique axis, i.e. the C screw rotation axis is along c . We use the experimental structural data for the structural parameters [6]. There are 2 inequivalent Mo’s and 4inequivalent Te’s, and they all occupy 2 e Wyckoff positions. Hence the multiplicity of the chemical formula units is4, and there are in total 12 atoms in the primitive cell. There are in total 56 valence electrons in the primitive unitcell, i.e. ν = 56. We list the HSPs, the irreps for each HSP, and the first principles calculated numbers n α i k i in Table I:According to Table I, we see that for HSPs Γ , B, Y, A , they all have four 1D irreps while the rest HSPs have one 2Dirrep. Thus there are in total 20 n k α s . Consider the filling number ν in the considered bands [1], the total numberof the entries for any BS in SG is 21. Form Table I, one can also readily find that the valence bands of β -MoTe constitute the BS as follows: n = (56 , , , , , , , , , , , , , , , , , , , ,
14) (4)= 12 a + 2 a + a + a + 12 a , (5) a b BiBr
FIG. 1. The crystal structures for β -MoTe (a) and BiBr (b).TABLE I. For SG , the HSPs are given by the labels Γ , B, . . . in order. For the labeling of the irreps of G ( k i ), we use ( j, m )where j means the j th irrep and m denotes the dimension of the corresponding irrep. They are all listed in Ref. [2]. We usethe same order of the irrep as Ref. [2]. The red color means that due to T , the irrep must occur with its T pair (belongingto the same irrep) simultaneously. Thus T requires that the red colored irreps must happen even times. So it is necessary todivide them by 2 [1] to obtain the physical common factors.HSP Γ B Y Z C D A E irrep (1,1) (2,1) (3,1) (4,1) (1,1) (2,1) (3,1) (4,1) (1,1) (2,1) (3,1) (4,1) (1,2) (1,2) (1,2) (1,1) (2,1) (3,1) (4,1) (1,2) n α i k i
16 16 12 12 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 where a has common factor 4, a , have common factor 2 while the others have no common factor as shown in Sec.XI. The expansion on the AI basis vectors is q = (12 , , , , ). The SI for β -MoTe is thus (0,0,2), and this indicatesa nontrivial topology of band [1].Based on the first principle calculations, we also obtain the parities of the valence bands, and list the numberof even(+)/odd( − ) parity in Table II A. Based on the obtained band parities, one can find that the conventional3D topological indices, i.e. ( ν ; ν , ν , ν ) [7], are all vanishing. However, the newly-introduced inversion topologicalinvariant κ [8, 11] is nonvanishing. It is defined by [8, 11]: κ = X k ∈ TRIM ( n + k − n − k ) / . (6) k ∈ TRIM Γ
X Y Z U T S Rn + k
16 14 14 14 14 14 14 14 n − k
12 14 14 14 14 14 14 14TABLE II. The calculated parities for the valence bands of β -MoTe . n ± k is the number of the occupied even/odd Kramerspairs (KPs), respectively. a bMoTe BiBr
FIG. 2. The comparison between the first principles calculated electronic bands (solid line) and the TB bands (dashed line)for (a) β -MoTe and (b) BiBr. It is easy to find that κ = 2 according to Table II A. Based on Eq. (6) and the Fu-Kane formula for ( ν ; ν , ν , ν )[7], one can easily understand the possible topological phase transition in β -MoTe and the phase diagram illustratedin Fig. 2(a) of the main text. Through adjusting the lattice structure by strain/pressure, one can tune the paritiesof the occupied bands. For example, a parity switch occurs at Γ point that the occupied bands are changed to own15 even KPs and 13 odd KPs. A topological transition to a strong topological insulator phase occurs. By this way, β -MoTe can further be tuned to be a trivial insulator by an additional parity switch. Thus for β -MoTe the possibletopological phase transition induced by strain/pressure is highly interesting.We construct a tight-binding (TB) model for β -MoTe in Sec. VII. This TB model not only reproduces the energybands reasonably shown in Fig. 2(a), it also gives exactly the same n α k s and mirror Chern numbers calculated by firstprinciples calculations. Based on the TB model we demonstrate the hinge states by constructing a centrosymmetricprism along c , and calculating the prism’s electronic structure. To unambiguously distinguish the hinge states fromthe bulk and surface states, we plotted the real space distributions of the eigen-states of the prism. In Fig. 3, weshow the real space distribution of two of the four hinge modes at the Fermi level (The other two are related to themby T ). These hinge states have their directions of spin locked to their motions forming a helical pattern as shown inFig. 2(c) in the main text. ab FIG. 3. The real space distribution for the two hinge states ψ , ψ at the Fermi level (with positive group velocities, See redlines in Fig. 2(d) of the main text for their dispersions) and the other two hinge states are related to them by T . m, n labelthe position while ρ mn ≡ P µ | ψ , ( m, n, µ ) | where summation is over µ , the collected set of the sublattice, orbital and spin. B. BiBr SG has a 2-fold screw axis which cannot protect any surface states [9] while for the neighbor SG , it hassymmorphic C rotation. This symmetry can protect surface Dirac cones in C symmetric surfaces [9]. We search SG , and we found another “2 in Z ” material: BiBr [10] with SG . We choose c as the unique axis (i.e. c is the C rotation axis). The fundamental lattice basis vectors can then be chosen as, a = 12 ( a − c ) , (7) a = b , (8) a = 12 ( a + c ) , (9)(10) a b FIG. 4. (a) The DOS plot for BiBr; (b) The BZ of BiBr and the (001)-surface BZ with the projection of the surface ¯ X and ¯ Y k point from the bulk momenta indicated by the dashed red line: note that we choose the primitive unit cell for the BZ ratherthan the Wigner-Seitz cell.TABLE III. For SG , the HSPs are given by the labels Γ , A, . . . in order, and their coordinates can be referred to Ref. [2].For the labeling of the irreps of G ( k i ), we use ( j, m ) where j means the j th irrep as listed in order by Ref. [2] and m denotesthe dimension. The red color means that due to T , the irrep must occur simultaneously with its T pair which belongs to thesame irrep.HSP Γ A Z M L V irrep (1,1) (2,1) (3,1) (4,1) (1,1) (2,1) (3,1) (4,1) (1,1) (2,1) (3,1) (4,1) (1,1) (2,1) (3,1) (4,1) (1,1) (2,1) (1,1) (2,1) n α i k i
18 18 14 14 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 where a , b , c are conventional unit cell vectors. There are four inequivalent Bi’s and four inequivalent Br’s, and theyall occupy the 4 i Wyckoff position. So the total number of atoms in the primitive unit cell is 16 (note that the Bravaisis of the base-centered type). The calculated BS for the 64 valence bands are given by, n = (64 , , , , , , , , , , , , , , , , , , , ,
16) (11)= 14 a + 2 a + 2 a + 2 a + a − a , (12)where a has common factor 4, a , have common factor 2 while the others have no common factor as shown in Sec.XI. The density of states (DOS) is plotted in Fig. 4, which shows that there is a full gap ( ∼ C protects surface Dirac cones. We construct a TB model as shown inSec. VII with its electronic energy spectrum shown in Fig. 2(b). It also reproduces the same BS and mirror Chernnumbers. Based on this model, we construct a slab and solve for the surface states. The slab geometry is finite along a with its infinite plane parallel to (001) (according to the conventional unit cell) plane. For this slab, the supercellunit vectors can be chosen as ¯a = a and ¯a = b . It is a oblique cell, whose convenient BZ can be chosen as theprimitive one rather than the Wigner-Seitz cell: the oblique tetragon spanned by, ¯b , ¯b : ¯b = 2 π ¯a × a a · ( ¯a × ¯a ) , (13) ¯b = − π ¯a × a a · ( ¯a × ¯a ) . (14)For each ¯k for this surface BZ, ¯k = k ¯b + k ¯b . The surface BZ is plotted in Fig. 4, with the projection from thebulk BZ indicated.For the calculation of the hinge states, we constructed a prism along c . The other two sides of this prism are chosenalong b and a , and they are both open. The same as β -MoTe , based on the TB model, we solve for the eigen-solutions for this prims. We distinguish the hinge states through the real space distributions of the eigen-states of theprism. C. AgXO,X=Na,K,Rb
Other than the centrosymmetric SG s , we also search those noncentrosymmetric ones but with S symmetry. Fo-cusing on SG , we found that a series of materials, AgXO with X=Na,K,Rb, are 1 in Z and the electronic bandplots for X=K and Rb are shown in Fig. 5. 1 in such Z indicates nontrivial topology and further, it corresponds toa 3D STI [8, 11]. For the centrosymmetric systems, one can use the Fu-Kane criteria [7] to judge a 3D topologicalphase. A similar criteria actually exists for the S symmetric systems [8, 11]. The corresponding topological invariantis defined by [8, 11]: κ = 12 √ X k ∈ SIM ,s s n s k mod 2 , (15)where SIM represents the set of S invariant momenta and s is the eigenvalue of S : s = e − i jzπ , j = ± , ± . Dueto T symmetry, s must occurs at the same time with s ∗ , though s ∗ maybe at a different k in the SIM. The SG here has a face-centered cubic lattice, whose four SIM are Γ , Z, P , P . They are listed in the Table IV, where wealso give the number of the occurrences for each eigenvalue s for the 18 valence bands at SIM by first principlescalculations. Note that the band inversion happens near Γ point, and when n ± Γ is changed to be 4 (and n ± Γ wouldbe 5), κ becomes zero and the materials become a trivial insulator. k ∈ SIM Γ
Z P Pn k n − k n k n − k n j z k is the number of the occupied eigen-states with S eigenvalue e − i jzπ . The coordinates of the four SIM are Γ = (0 , , , Z = (0 , , , P = (0 , , − ) , P = (0 , , ) respectively inthe conventional and reciprocal basis vectors. D. k · p model for the Dirac semimetal in the main text For the Dirac semimetal Bi MgO , we derive the low energy k · p Hamiltonian as follows: Consider the Hamiltonian H ( k ) in Γ- Z line, it is subject to the symmetry restriction of C z and σ v . We choose σ v is perpendicular to y axis. AgKOAgRbO ab FIG. 5. The electronic band plots for two noncentrosymmetric strong topological insulators: (a) AgKO and (b) AgRbO.
The band inversion occurs in Γ- Z between two bands near the Fermi level shown in Fig. 4 of the main text. The twobands belong to Λ and Λ irreps respectively of the symmetry group C v . For the Λ band, the two basis vectors are | j z > ( j z = ± ) while for the Λ band, the two basis vectors are | j z > with j z = ± . j z means that the eigenvalueof C z is e − i jzπ . σ v relate | j z > to | − j z > . So each band is two-fold degenerate, which is also dedicated by thejoint operation T I . Hence the representation matrix for C z is dia( e − i π , e i π , e − i π , e i π ) and for σ v , it is iσ ⊗ σ (note σ v = − and Λ band representations, H ( k ) = M ( k ) σ ⊗ σ . So in Γ- Z , when M ( k ) = 0 has a solution k = k D , a band crossing happens at this point. The low energy Hamiltonian to the linearterm around this point is written as h ( q ) = ∂H ( k D ) ∂ k D · q ≡ H (1) q . h ( q ) is restricted by C z and σ v thatD( C z ) † H (1) D( C z ) = C z H (1) , (16)D( σ v ) † H (1) D( σ v ) = σ v H (1) . (17) T I = U K also has effect to the k · p Hamiltonian by ( U † H (1) U ) ∗ = H (1) where U = σ ⊗ σ . We finally obtain thesymmetry allowed k · p Hamiltonian as follows (in the basis of {| >, | >, | − >, | − > } ): h ( q ) = (cid:18) c q σ + c q σ + c ( q σ + q σ ) 00 c q σ + c q σ + c ( q σ − q σ ) (cid:19) , (18)where the two 2 × q z dependentconstant term. III. THE OTHER “2 IN Z ” MATERIALSA. Hourglass insulator TiS As shown in BiBr of the main text, symmetry other than inversion may protect gapless surface states. In that case, C rotation plays such a role while the screw 2-fold axis in MoTe cannot. However the other kind of nonsymmorphicoperation, i.e. glide plane may protect novel hourglass surface states [12]. We thus search the SG which hasmore symmetry operations especially some glide planes and we found that the cubic TiS (c-TiS ) [13], is 2 in Z .For SG , there are 8 AI basis vectors while only one has common factor (equal to 4) [1] (shown in Sec. XI),thus its SI group X BS = Z . For c-TiS , a primitive unit cell contributes ν = 64 valence electrons (For Ti and S,consider 4 and 6 valence electrons respectively while the multiplicity of the chemical formula units in the primitiveunit cell is 4). Following the SI strategy, we calculate n α k for HSPs up to the 64th band. All these numbers areintegers. In Fig. 6,there is actually a small but finite direct gap at Γ point. The expansion on the AI basis vectors is q = (3 , − , , , , , , − ) . Thus n α k s can constitute a BS, and furthermore this BS has a nontrivial SI (i.e. it has 2in the Z group). Parity calculations show that ( ν ; ν , ν , ν ) are all vanishing. Thus c-TiS is not a strong or weakTI. The inversion topological invariant [8, 11] i.e. κ is equal to 2. This means there is opportunity that we observethe gapless hinge states in the domain wall of two gapped side surfaces. We note that c-TiS has several glide planes,thus may possesses novel surface states. Here the hourglass index ( δ g protected by glide plane (001)) must be 1[11] thus we expect hourglass surface states to appear in the surface termination where the above glide symmetry ispreserved. FIG. 6. The band plot for the hourglass topological insulator c-TiS . Note that at Γ point, there is a finite direct gap. B. Weak topological insulator: Ag F Even with only inversion symmetry, namely SG , it can also protect gapless surface states for the compounds withthe SI as 2 in the key Z . Ag F is such kind of material whose n α k s can constitute a BS indicating the existenceof a continuous direct gap throughout the BZ. Its SI is found to be (1 , , ,
2) ( X BS = Z × Z × Z × Z [1]). Asthe Z factors in X BS here corresponds to weak TIs [7], this material is a weak TI with translation protected gaplesssurface states appearing in (010) and (001) surface. For its 228 valence bands, our first principles calculations showthat apart from Γ and X points, these 114 KPs at each of the rest TRIM are classified to 57 even and 57 odd KPs.For Γ and X points, they both have 66 even and 48 odd KPs. Thus κ = 2, and it hosts 1D hinge states. Besides, ν = 1 and ν , , = 0. By tuning the occupied band parities at Γ to 68 even KPs and 46 odd KPs, this compoundwill become the “0 in Z ” phase as shown in Fig. 2(a). κ is then equal to 0. So the 1D hinge states will disappear.However ν is still 1, thus the topological surface states still exist [7]. Seen from the electronic structure plot shownin Fig. 7, although its full gap is not very large, the dispersion looks very clear and so the projection of the bulkbands to the surface BZ can demonstrate several empty zones for the accommodation of the surface states. This isvery favorable for the experimental observations. FIG. 7. The energy band plot for the weak TI for Ag F . C. Helical Hinge gapless states in A7 phosphorus
It is well-known that the elementary phosphorus (P) owns many kinds of allotropes such as the Black phosphorus[14], a famous layer material. Under pressure about 9 GPa (whose band plot is shown Fig. 8 ), P crystallizes in SG (A7 phase) [15]. For this SG , it has in total 8 AI basis vectors while only two have common factor, one is 4while the other is 2, thus its SI group is X BS = Z × Z [1]. For the A7 P, it has 10 valence electrons in the primitiveunit cell, and we calculate the n α k s for the first 10 bands. The expansion coefficients are (0,0,1,-1,1,0,-1, − ) (SeeTable for the corresponding AI basis vectors), thus the A7 P, is a BI which has a finite direct band gap everywhere inthe BZ and has a nonvanishing SI=(0,2). Furthermore its inversion topological invariant is κ = 2 [8, 11], thus it mustbe a topologically nontrivial insulator although all the conventional topological invariants are found to be vanishing.1It displays inversion-protected gapless hinge states as long as the open conditions preserve the inversion symmetry.We note that recently there has been an experimental observation of such hinge states on bismuth [16] which sharesthe same crystal structure and SI as A7 P here. FIG. 8. The energy band plot of the A7-P.
IV. PREDICTED STRONG TOPOLOGICAL INSULATORS BY , IN Z We also find several materials having “1 (or 3) in Z ” which must be strong TIs. Their electronic band plots aregathered in Fig. 9. These materials all own finite direct gap everywhere in the whole BZ, and for Bi PbTe , it has afull gap ( ∼
66 meV).
TABLE V. Table of noncentrosymmetric STI candidates discovered by 1 , Z . SG Material X BS SI ( ν ; ν , ν , ν ) κ CaAs [17] Z × Z × Z × Z (0,0,1,1) (1;1,0,0) 1 Bi PbTe [18] Z × Z (1,1) (1;1,1,1) 3 CaGa As [19] Z × Z (1,1) (1;1,1,1) 1 V. THE OTHER PREDICTED SEMIMETALS BY SI METHOD
Through the expansion coefficients q i s , we found another two topological (semi-)metals in this sections: three-folddegenerate fermions in AuLiMgSn and nodal-line semimetal AgF .2 CaAs CaGa As Bi PbTe a bc FIG. 9. The band plots for all the STIs found by “1 (or 3) in Z ”. A. Three-fold degenerate fermions
The three-fold degenerate fermions are found for AuLiMgSn [20] which has SG n α k s are allintegers thus there are finite direct gaps in all the HSPs. However expansion on the SG s AI basis vectors showsthat they cannot constitute a BS at all, namely case 3 in the main text. Thus there must be some band crossing.Based on the first principles calculations , we find that in the symmetry line Γ L , the Γ and Γ bands shown in Fig. 10cross with each other, resulting in a three-fold degenerate fermion (TF). Meanwhile near such a band crossing, thereis also another TF originated from the crossing between Γ and Γ . These TFs are protected by C v symmetry alongthe Γ L . Furthermore, in the symmetry line Γ K whose symmetry group is S , we also observe two band crossingsbetween two nondegenerate bands, i.e. resulting in two Weyl points (WPs). These WPs carry a zero topologicalcharge due to S . They are also shown in Fig. 10. B. The nodal-line semimetal AgF AgF [21]( SG ) is predicted as a topological bulk hourglass nodal-line semimetal. In AgF , there are in total 60valence electrons. Thus ν e = 60. We calculate the numbers n α k s up the 60th bands and find that they are all integers,which means that at least at HSPs the valence bands are gapped from the conduction bands. However they cannot3 TPs WPs
FIG. 10. We show the band plot for the AuLiMgSn. Only Γ in the Γ L line are 2D irrep and the others are all 1D. Thusin Γ L line, the Γ band crosses with the Γ and Γ , resulting in two kinds of three-fold degenerate points (TPs) or three-folddegenerate fermions, protected by C v . In Γ K line, Γ and Γ crosses with each other resulting in two type-II Weyl points(WPs), which are protected by S symmetry. constitute a BS at all because the expansion is ( , , ) (See Sec. XI for the AI basis vectors of SG ). Hence theremust be some band crossing(s) in the BZ. Inspecting all the high symmetry lines and planes, we find that in k x = πa plane, there is a large four-fold-degenerate nodal-line shown in Fig. 11(c) by first principles calculations. The glidesymmetry, i.e. ˜ M x = ( − x + , y + , z ), will guarantee that any curve connecting S and P (arbitrary point in U X )will possess an unavoidable hourglass type band crossing (See Fig. 11(a)). The crossing point is robust and protectedby ˜ M x because it is originated from two 2-fold degenerate bands with inverse eigenvalue of ˜ M x . These crossing pointsform a hourglass Dirac nodal line in Fig. 11(c).The SG is a nonsymmorphic group with two glides: ˜ M x = ( − x + , y + , z ), ˜ M y = ( x, − y + , z + ) andinversion I . A third glide plane can be obtained by the product of the above two: ˜ M z = ( x + , y, − z + ). Inthe U - X line i.e. k x = π, k y = 0, the symmetry includes ˜ M x and ˜ M y , T I and their products. We first considerthe first two operations. The eigenvalues of them can be quickly obtained through: ˜ M x = { ¯ E | , , } = − M y = { ¯ E | , , } = − e − ik z where ¯ E represent spin-2 π rotation. Thus the eigenvalues for the two can be g x = ± i and g y = ± ie − i kz . Besides they anticommutates with each other, therefore we can only use g x or g y to label theBloch eigen-states. We choose g x here and label the Bloch states as | UX , g x > , and ˜ M y | UX , g x > = e − ik z | UX , − g x > .Hence | UX , g x > and | UX , − g x > will share the same eigen energy (they are orthogonal to each other because of theinverse g x ).It is well-known that T I enforces each band to be at least two-fold degenerate. And because I , ˜ M x = 0, T I willpreserve the eigenvalue g x , i.e., ˜ M x ( T I| UX , g x > ) = g x ( T I| UX , g x > ). This will require the aforementioned doubletto be doubled, which results a 4D irrep [2].We then consider the symmetry line S - X , i.e. k x = π, k z = 0. Then we should consider ˜ M x , ˜ M z , T I and theirproducts. As before, we first consider the eigen-values of the first two operations in this line. Because ˜ M x = − e − ik y M z = 1, then the eigenvalues have to be g x = ± ie − i ky and g z = ± | SX , g x , g z > .Actually there are 4 1D irrep’s in this line [2], UX RSPb ca
FIG. 11. (a) shows the energy band plot of AgF . (b) depicts the Brillouin zone in the left panel. In the up right panel of(b), from S to any point P in UX , there is always a hourglass type band crossing. This then results a nodal line as shown inthe down right panel of (b) based on the first principles calculations. Again we consider
T I . First ˜ M x I = { E | , , }I ˜ M x = − e − ik y ˜ M x . This means that T I will preserve g x . Similarly, T I will reverse g z . Hence, in S - X , two-fold (due to T I ) degenerate bands bear the same g x and inverse g z . In anotherwords in S - X , there are two different 2D irreps when considering time reversal symmetry, which will converge into a 4Dirrep at S or two 4D irrep’s at X [2]. Thus it is possible for the band crossing to happen in S - X . S and X are both time-reversal invariant. And P commutates or anti-commutates with ˜ M x for S or X respectively, while P anti-commutateswith ˜ M z both for S and X . At S , P| S , g x , g z > bears the same eigenvalue g x while inverse eigenvalue g z as | S , g x , g z > . T will not change both g x and g z , thus at S the quartet | S , g x , g z >, I| S , g x , g z >, T | S , g x , g z >, T I| S , g x , g z > bearsthe same g x and are mutually orthogonal. Actually S bears two 2D irrep’s both are doubled by T [2]. However, at X ,it is similar to find that | X , g x , g z >, I| X , g x , g z >, T | X , g x , g z >, T I| X , g x , g z > have g x = ( i, − i, − i, i ) or ( − i, i, i, − i ).Because the eigenvalue g x continuously changes by g x = ± ie − i ky , there must be states switch between the quartetsof S and X through S - X , which results in a hourglass pattern and a Dirac point. Further as g x is well-defined in thewhole line U - X , any curve connecting S and one point in U - X would give a hourglass type Dirac band crossing. This5is verified by our first principles calculations shown in Fig. 11(b). This is why we cannot obtain a BS when considerbands up to the filling. VI. FIRST PRINCIPLES CALCULATIONS OF MIRROR CHERN NUMBERSA. Techniques involved in first principles calculations of mirror Chern numbers
The central task to calculate the (mirror) Chern number is to calculate the overlap matrix S nn ( k , k ) = where the inner product is the integration in the primitive unit cell and u n k is the Bloch eigen-state.The mirror Chern number, namely C M is defined in the mirror symmetric plane in the BZ: C M = ( C + iM − C − iM ),where the superscript ± i labels the eigenvalues of the mirror operation. Due to T , C + iM = − C − iM , thus C M = C + iM .By definition: C ± iM = X n ∈ occ. Z BZ d k Ω ± i ( n k ) , (19)where occ. denotes the occupied bands and Ω ± i ( n k ) is the Berry curvature at k for | u ± in k > where we have used themirror eigenvalues to label the lattice-periodic function | u ± in k > :Ω ± i ( n k ) = ( ∇ × A ± i ( n k )) ⊥ , (20)where ⊥ means the direction perpendicular to the symmetric plane, and A is the Berry connection: A ± i ( n k ) = i . For a small portion ∆ S is the symmetry plane, the following relation must hold: γ ± i ∆ S ( n ) = Z ∆ S d k Ω ± i ( n k ) = I ∂S d r · A ± i ( n k ) ∈ ( − π, π ] , (21)where ∂S represent the boundary of ∆ S and we have chosen a gauge that the Berry phase is restricted into ( π, − π ]which is required by that the Berry curvature is finite while ∆ S is very small. Because, A ± i ( n k ) = i < u ± i ( n k ) | u ± i ( n, k + δ k ) > − < u ± i ( n k ) | u ± i ( n, k ) >δ k , (22)we will have, < u ± i ( n k ) | u ± i ( n, k + δ k ) > = e − i A ± i ( n k ) · δ k . (23)Thus we can divide the loop ∂S into several parts: ( k , k ] ∪ ( k , k ] ∪ . . . ∪ ( k N − , k N − ] ∪ ( k N − , k ]. According toEq. (23), γ ± i ∆ S ( n ) = − Im log ( < u ± in k | u ± in k >< u ± in k | u ± in k > . . . < u ± in k N − | u ± in k > ) . (24)It is then easy to generalize to the multi-band case: X n ∈ occ. γ ± i ∆ S ( n ) = − Im log det[ S ± i ( k , k ) S ± i ( k , k ) . . . S ± i ( k N − , k )] , (25)where det is short for determinant , S is the overlap matrix whose row and column indices are the occupied bandindices n, n , S is ν × ν assuming there are in total ν occupied bands and the superscripts ± i labels the eigenvalueof mirror as before. Eq. (25) is clearly gauge independent for the simultaneous presences in the bra and ket for aeigen-function. Note that the first principles eigen-functions generally are not simultaneously the eigenstates of themirror operator, thus a transformation should be made to obtain the eigenstates of the mirror, denoted as U m ( k ) as k with the first half with eigenvalue + i and the rest with eigenvalue − i . Write Eq. (25) as follows: X n ∈ occ. γ ∆ S ( n ) = − Im log | S ( k , k ) S ( k , k ) . . . S ( k N − , k ) | (26)= − Im log | U m ( k ) † S ( k , k ) S ( k , k ) . . . S ( k N − , k ) U m ( k ) | (27)= − Im log | [ U m ( k ) † S ( k , k ) S ( k , k ) . . . S ( k N − , k ) U m ( k )] ∼ ν/ , ∼ ν/ | (28) − Im log | [ U m ( k ) † S ( k , k ) S ( k , k ) . . . S ( k N − , k ) U m ( k )] ν/ ∼ ν,ν/ ∼ ν | , (29)where in the last equality, the two parts correspond to P n γ ∆ S ( n ) + i and P n γ ∆ S ( n ) − i respectively. Finally, tocalculate the mirror Chern number, we just need to calculate the overlap matrix S ( k , k ) no matter whether theeigenfunctions involved are the eigen-states of mirror or not and we just need to make a unitary transformation shownabove to extract the parts for each mirror-eigenvalue.6 FIG. 12. The sketch for numerical calculation of the Chern number. First we dived the 2D BZ into N × N small portions. Foreach portion ∆ S , we then calculate the Berry phase around its boundary ∂S through further dividing it into several segments(red dots). B. Details for calculation of C M in β -MoTe and BiBr For β -MoTe , the mirror plane is perpendicular to the z (or c ) direction: M z = ( x, y, − z + ), we should focus ontwo mirror symmetric planes k z = 0 and k z = πc respectively. Note that in k z = πc plane, T I will preserve the mirroreigen-values due to the c translation in M z : i.e. when ψ is the eigenstate of M z , T I ψ will have the same eigenvalueof M z . Not that T I will change the sign of the Berry curvature, thus the mirror Chern will be vanishing for the k z = πc plane. For the k z = 0 plane, we exploit the above technique through linearized augmented plane-wave methodas implemented in WIEN2k package [22]. We divide the k z = 0 plane to to 50 ×
50 parts and for each part we calculatethe Berry phase around them (Eq. (26)) by dividing each part into 8 portions (schematically shown in Fig .12). Thecalculated MCNs are found to be vanishing. According to the formula of C + M = i P n R d k < ∂ k x u + n | ∂ k y u + n > − c.c. where + represents that the eigenvalue of the mirror operator is + i , n is the occupied band index and the integralzone is restricted to a mirror symmetric plane. Our calculations find that the eigen-state u + n has a weak dependenceon k x which is consistent with that the MCNs are all vanishing.For BiBr, the mirror plane is also perpendicular to c in our adopted setting. In this case, T I will reverse theeigenvalue of the mirror operator. So we should calculate the mirror Chern numbers for both k z = 0 and k z = πc planes. We take the similar way of partitioning the 2D BZ as β -MoTe .7 VII. TIGHT BINDING MODEL FOR MoTe AND BiBr
For the calculations of the hinge states for MoTe and BiBr, we need to take open conditions in two directions. Itwould be rather computationally demanding especially for the first principles calculation. Thus we construct a tightbinding (TB) model for both materials:ˆ H T B = X R , R ; s,s ; µ,µ ; σ,σ h ( R , R ; s, s ; µ, µ ; σ, σ ) ˆ C † R + τ s ,µ,σ ˆ C R + τ s ,µ ,σ , (30)where R , R label the primitive unit cell within which the atoms are located at τ s , τ s relative the primitive unit cell. µ, µ label the orbital degree of freedom and σ, σ label the spin eigenvalue of S z (the z component of the spin operator S ). ˆ C † ( ˆ C ) is the creation (annihilation) operator of the state as denoted in its subscript. We adopt orthogonal atomicorbitals. In each atom, we choose appropriate atomic orbitals which dominate the contributions near the Fermi level.Symmetry (time-reversal and space group) imposes restrictions for the Hamiltonian matrix elements. We thus takethe Slater-Koster (SK) formalism wherein the hopping integrals are given by several adjustable parameters and thehopping direction cosines. While for the onsite terms, the crystal field splitting can be described by the site-symmetryallowed onsite Hamiltonian matrix. The spin-orbit coupling (SOC) is given by,ˆ H SO = X R s,µν,σσ ( λ s L · S ) µσ,νσ ˆ C † R + τ s ,µ,σ ˆ C R + τ s ,ν,σ , (31)where L is the orbital momentum and λ s is the SOC parameter: for those atoms related by SG symmetry, they sharethe same SOC parameter while note that λ s s also take a different values for different l s .With the above TB model at hand, we then fit the energy bands from the first principles calculation near the Fermilevel within the irreducible BZ for both materials, by least squares method to obtain a optimized parameters. Thecomparisons of the TB electronic bands with the first principles results in Fig. 2 show that for both materials, theTB models reasonably reproduce the first principles bands. Besides the energy bands, it is also required to reproduceexactly the same SI and mirror Chern numbers as the first principles results. VIII. METHOD
The electronic band structure calculations have been carried out using the full potential linearized augmentedplane-wave method as implemented in the WIEN2K package [22]. The generalized gradient approximation (GGA)with Perdew-Burke-Ernzerhof (PBE) [23] realization was adopted for the exchange-correlation functional. It is worthpointing out that the modified Becke-Johnson exchange potential for the correlation potential (MBJ) [24] has alsobeen used and we found that it has no affect on our main results.
IX. CALCULATION OF ATOMIC INSULATOR BASIS
The AI basis vectors are the central objects of our SI method for screen the materials database to search fortopological materials. Step by step we show how to determine the AI basis for a SG , taking SG as the example: Step 1 : Obtaining HSPs.
For SG , the HSPs are just 8 time-reversal invariant momenta (TRIM): k = ( k , k , k ) written inthe basis of the reciprocal lattice basis vectors and k i takes 0 or shown in Table VI. The irreps for each HSP. SG s HSP Γ
X Y Z U T S R coordinate (0,0,0) ( , ,
0) (0 , ,
0) (0 , , ) ( , ,
0) (0 , , ) ( , , ) ( , , )TABLE VI. The coordinates of eight high symmetry points (HSPs) for SG For each high symmetry point k , the little group G ( k ) is also SG containing only two different irreps: D ( k ) and D ( k ) that: { p | R } ψ α k ,i = X i D α k ,i i ( { p | R } ) ψ α k ,i , (32)8where { p | R } ∈ G ( k ), D α k ,i i ( { p | R + R i b · R D α k ,i i ( { p | R } ), α labels the irrep taking 1 or 2, and i, i denote the basisvector for the irrep.To clearly demonstrate an irrep, we will use the subscript i ( i = 1 , , . . . , r , r is the total number of the irreps)added to the name of the HSP to label the i th irrep of this HSP. Note that some irrep can be doubled by thetime-reversal operation ( T ), which means that this irrep must occur even times, then we should divide the correspondnumber n α k by 2[1]. In this case, the irrep is in red. The dimension of the irrep will be denoted by the superscriptadded to the HSP’s name, e.g. Γ , Γ , Γ , Γ , . . . represents the irreps at Γ (we will demonstrate the irreps hereafterfollowing the same order shown in Ref. [2]): Γ and Γ are both 1D irreps while Γ is 1D irrep doubled by T and Γ is 2D irrep.For SG here, D α ( I ) = ( − α − , i.e. α = 1 (the first irrep) corresponds to the states with even parity while α = 2(the second irrep) corresponds to the states with odd parity. That is to say, every HSP of SG has two 1D irrep(each one is doubled by T ). So the total number of n α k s is 16. Consider the filling number ν , there are 17 entriesin any BS with SG . Consider eight compatibility relations ν = n k + n k . Then only 9 entries are independent, i.e. d BS = 9. For an arbitrary SG , the compatibility relations can get much more complex. However instead of directlyanalyzing the compatibility relations, we can detour to obtaining the AI basis vectors [1]. Any AI, itself a BS, willautomatically satisfy the requirement of the compatibility relations. To obtain the AI basis vectors, one need toexhaustively consider all the Wyckoff positions and all the site-symmetry irreps. Step 2 : Give all the Wyckoff positions.
The Wykcoff positions for SG is shown in Table VII. SG s Wyckoff position site-group Wyckoff orbits2i C ( x, y, z ) , ( − x, − y, − z )1h C i (1 / , / , / C i (0 , / , / C i (1 / , , / C i (1 / , / , C i (1 / , , C i (0 , / , C i (0 , , / C i (0 , , SG . List all the site-symmetry irreps for every Wyckoff position.
Given a Wyckoff position m W (like 2 a, b, . . . , m counts the number of sites in this Wyckoff orbit), its site symmetry group is then determined. Writing the sitesas { r W , r W , . . . , r W m } in one primitive unit cell with the operations { p | R } in the SG satisfying { p | R } r W = r W constituting the site symmetry group. The number of the sites in the Wyckoff orbit m = |G ||G ( r W ) | ( G is the pointgroup), and there must exist a SG element which will give r W J ( J = 1 , , . . . , m ) from r W : { p J | R J } r W = r W J . Denotingthe basis functions for one irrep ( D βs of the site symmetry group as φ β r W ,j , i.e., for { p | R } ∈ G ( r W ) , { p | R } φ β r W ,j =D β r W ,j j ( { p | R } ) φ β r W ,j . Note that φ β r W ,j may not be localized on r W . Then we can obtain other real-space orbitals: { p J | R J + R } φ β r W ,j , which constitute a complete basis for the SG . Through Bloch summation: ψ β k ,J,j = X R e i k · ( r W J + R ) { p J | R J } φ β r W ,j , (33)which is the basis for some k point in the BZ. By the technique shown in Sec. X, one can obtain all the n α k s for theAI corresponding to this Wyckoff position and site-symmetry irrep. C ED s D s for C considering T . Note we only consider the doubled-valued irreps, thus the character wouldbe multiplied by − π more rotation. Step 3 : Make Smith normal decomposition.
According to the Wyckoff positions listed in Table VII, and usingthe corresponding site-group irreps shown in Tables VIII and IX, we thus obtain 17 AIs, denoted by n im W with the9 C i E PD s D s − D s for C i considering T . SG i n h n h n g n g n f n f n e n e n d n d n c n c n b n b n a n a ν X X Y Y Z Z U U T T S S R R SG superscript labeling the site-group-irrep in the same order shown in Tables VIII and IX, and the subscript labelsthe Wyckoff position. They are listed in Table X. These 17 vectors are generally not independent with each other,we can make a so-called Smith normal decomposition which will give d AI ( ≤
17) AI basis vectors through the linearcombination of these 17 AI vectors. The results are shown in Table X(a) where the 9 basis vectors are printed explicitlyand they are in an ascending order of the common factors. So d SG AI = 9 which is equal to d SG BS . Ref. 1 proved that itholds for each of 230 SG s [1]. It is easy to find that a , . . . , a have no common factors while a , a , a have a commonfactor (=2), and a have a common factor (=4). Thus the symmetry indicator (SI) group is X BS = Z × Z × Z × Z .Further more we can find that the condition for the filling of a band insulator is 2 N . X. REDUCTION OF AI ON HSPS
Focused on HSPs, we can then easily obtain the number of occurrences for the irrep α of G ( k ): n α k for the abovebasis (generally reducible) by: n α k = 1 |G ( k ) | X g ∈G ( k ) χ α k ( g ) ∗ χ k ( g ) , (34)where χ denotes the character. χ k ( g ) is the character for the basis in Eq. (33), can be obtained through ( g ∈ G ( k )): gψ β k ,J,j = X J ,j D J ,j ; J,j ( g ) ψ β k ,J ,j , (35)where χ k = tr (D). The D is obtained by first knowing the permutation (i.e. J → J ) of the atoms: g { p J | R J } = { p J | R J + R } g , (36)where g ∈ G ( r W ), thus, D J ,j ; J,j ( g ) = e i k · ( r W J − r W J ) − i k · R D β r W ,j j ( g ) . (37)Note that when we only need to calculate the trace of the representation matrix D( g ), we can just consider J = J butfor some cases we should know exactly the representation matrix, e.g. when calculating the mirror Chern number, weshould know the representation matrix for the mirror operation.0 XI. ATOMIC BASIS VECTORS
In this section, we list the AI basis vectors for all the SG s we encounter in this work. (a)The 9 AI basis vectors for SG . Here ν is the number of the bands. It is alsocalled the filling number. Staring fromthe 3rd row, we give the number n α k inorder. We omit the notation n forclarity: The first column of these rowsgives the information of the HSP andits irrep completely. SG a a a a a a a a ν X X Y Y Z Z U U T T S S R R (b)The 5 AI basis vectorsfor SG . SG
11 a a a a a ν B B B B Y Y Y Y Z C D A A A A E (c)The 7 AI basis vectors for SG . SG
12 a a a a a a a ν A A A A Z Z Z Z M M M M L L V V , 50 (2017).[2] C. J. Bradley and A. P. Cracknell, The Mathematical Theory of Symmetry in Solids: Representation Theory for PointGroups and Space Groups (Oxford University Press, 1972).[3] H. C. Po, H. Watanabe, and A. Vishwanath, ArXiv e-prints (2017), arXiv:1709.06551.[4] H. Watanabe, H. C. Po, and A. Vishwanath, arXiv preprint arXiv:1707.01903 (2017).[5] C. Fang, Z. Song, and T. Zhang, ArXiv e-prints (2017), arXiv:1711.11050.[6] B. E. Brown, Acta Crystallographica , 268 (1966).[7] L. Fu and C. L. Kane, Phys. Rev. B , 045302 (2007).[8] E. Khalaf, H. C. Po, A. Vishwanath, and H. Watanabe, ArXiv e-prints (2017), arXiv:1711.11589.[9] C. Fang and L. Fu, arXiv preprint arXiv:1709.01929 (2017).[10] von Benda Heike, S. Arndt, and B. Wolfgang, Zeitschrift f¨ur anorganische und allgemeine Chemie , 53 (1978).[11] Z. Song, T. Zhang, Z. Fang, and C. Fang, ArXiv e-prints (2017), arXiv:1711.11049.[12] Z. Wang, A. Alexandradinata, R. J. Cava, and B. A. Bernevig, Nature , 189 EP (2016), article.[13] P. Lightfoot, F. Krok, J. Nowinski, and P. Bruce, Journal of Materials Chemistry , 139 (1992).[14] L. Li, Y. Yu, G. Ye, Q. Ge, H. Ou, X.D.and Wu, D. Feng, X. Chen, and Y. Zhang, NATURE NANOTECHNOLOGY ,372 (2014).[15] T. Kikegawa and H. Iwasaki, Acta Crystallographica Section B-Structral Science , 158 (1983).[16] F. Schindler, Z. Wang, M. G. Vergniory, A. M. Cook, A. Murani, S. Sengupta, A. Y. Kasumov, R. Deblock, S. Jeon,I. Drozdov, H. Bouchiat, S. Gu´eron, A. Yazdani, B. A. Bernevig, and T. Neupert, ArXiv e-prints (2018), arXiv:1802.02585.[17] W. Bauhofer, M. Wittmann, and H. von Schnering, Journal of Physics and Chemistry of Solids , 687 (1981).[18] R. Chami, G. Brun, J. Tedenac, and M. Maurin, Revue de Chimie Minerale , 305 (1983).[19] H. Hua, R. Stearrett, E. Nowak, and S. Bobev, European Journal of Inorganic Chemistry (online) , 4025 (2011). (d)The 3 AI basisvectors for SG . SG
61 a a a ν
16 8 -32Γ Y Y X X Z Z U U T T S S R R R R R R R R (e)The 8 AI basis vectors for SG . SG
136 a a a a a ν K K K K K K K K K K K K K K K K , 1341 (1980).[21] P. Fischer, D. Schwarzenbach, and H. Rietveld, Journal of Physics and Chemistry of Solids , 543 (1971).[22] P. Blaha, K. Schwarz, G. Madsen, D. Kvasicka, and J. Luitz, An Augmented Plane Wave Plus Local Orbitals Program forCalculating Crystal Properties (2001).[23] B. K. Perdew, J. P. and M. Ernzerhof, Phys, Rev. Lett. , 3865 (1996).[24] F. Tran and P. Blaha, Phys, Rev. Lett. , 226401 (2009). (f)The 8 AI basis vectors for SG . SG
166 a a a a a a a a ν
12 6 6 -4 0 2 -4 8Γ Z Z Z Z Z Z L L L L F F F F (g)The 3 AI basis vectors for SG . SG
216 a a a a a a ν
12 4 2 2 2 -4Γ X X L L L W W W W (h)The 8 AI basis vectors for SG . SG
227 a a a a a a a a ν
24 8 8 8 -24 8 4 -48Γ X L L L L L L W W W W W25