Topological quantum chemistry
Barry Bradlyn, L. Elcoro, Jennifer Cano, M. G. Vergniory, Zhijun Wang, C. Felser, M. I. Aroyo, B. Andrei Bernevig
TTopological Quantum Chemistry
Barry Bradlyn, ∗ L. Elcoro, ∗ Jennifer Cano, ∗ M. G. Vergniory,
3, 4, 5, ∗ Zhijun Wang, ∗ C. Felser, M. I. Aroyo, and B. Andrei Bernevig
6, 3, 8, 9, † Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA Department of Condensed Matter Physics, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain Donostia International Physics Center, P. Manuel de Lardizabal 4, 20018 Donostia-San Sebasti´an, Spain Department of Applied Physics II, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain Max Planck Institute for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany. Department of Physics, Princeton University, Princeton, New Jersey 08544, USA Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany Laboratoire Pierre Aigrain, Ecole Normale Sup´erieure-PSL Research University,CNRS, Universit´e Pierre et Marie Curie-Sorbonne Universit´es,Universit´e Paris Diderot-Sorbonne Paris Cit´e, 24 rue Lhomond, 75231 Paris Cedex 05, France Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7589, LPTHE, F-75005, Paris, France (Dated: July 21, 2017)The past decade’s apparent success in predicting and experimentally discovering distinct classesof topological insulators (TIs) and semimetals masks a fundamental shortcoming: out of 200,000stoichiometric compounds extant in material databases, only several hundred of them are topologi-cally nontrivial. Are TIs that esoteric, or does this reflect a fundamental problem with the currentpiecemeal approach to finding them? To address this, we propose a new and complete electronicband theory that highlights the link between topology and local chemical bonding, and combinesthis with the conventional band theory of electrons. Topological Quantum Chemistry is a descrip-tion of the universal global properties of all possible band structures and materials, comprised ofa graph theoretical description of momentum space and a dual group theoretical description in realspace. We classify the possible band structures for all
230 crystal symmetry groups that arise fromlocal atomic orbitals, and show which are topologically nontrivial. We show how our topologicalband theory sheds new light on known TIs, and demonstrate the power of our method to predict aplethora of new TIs.
I. INTRODUCTION
For the past century, chemists and physicists have advocated fundamentally different perspectives on materials:while chemists have adopted an intuitive “local” viewpoint of hybridization, ionic chemical bonding and finite-rangeinteractions, physicists have described materials through band-structures or Fermi surfaces in a nonlocal, momentum-space picture. These two descriptions seem disjoint, especially with the advent of TIs, which are exclusively understoodin terms of the nontrivial topology of Bloch Hamiltonians throughout the Brillouin zone (BZ) in momentum space.Despite the apparent success that the field has had in predicting some [mostly time-reversal (TR) invariant] TIs,conventional band theory is ill-suited to a natural treatment of TIs. Given the paucity of known TIs (less than 400materials out of 200,000 existent in crystal structure databases!), one may ask whether topological materials are trulyso rare, or if this reflects a failing of the conventional theory.By their very nature, the topological properties of energy bands are properties global in momentum space. Theduality between real and momentum (direct and reciprocal) space suggests that properties of bands which are nonlocalin momentum space will manifest locally in real space. In this paper, we unify the real and momentum spacedescriptions of solids, and in doing so provide a new, powerful, complete and predictive band theory. Our procedureprovides a complete understanding of the structure of bands in a material and links its topological to the chemicalorbitals at the Fermi level. It is therefore a theory of Topological Quantum Chemistry.Developing a complete theory of topological bands requires an extremely large body of work made up of severalingredients. First, we compile all the possible ways energy bands in a solid can be connected throughout the BZ toobtain all realizable band structures in all non-magnetic space-groups. Crystal symmetries place strong constraints onthe allowed connections of bands. At high symmetry points, k i , in the BZ, Bloch functions are classified by irreduciblerepresentations (irreps) of the symmetry group of k i , which also determine the degeneracy. Away from these high ∗ These authors contributed equally to the preparation of this work. † To Whom correspondence should be addressed; Permanent Address: Department of Physics, Princeton University, Princeton, NewJersey 08544, USA a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l symmetry points, fewer symmetry constraints exist, and some degeneracies are lowered. This is the heart of the k · p approach[1] to band structure, which gives a good description nearby high symmetry k − points. However, the globalband structure requires patching together the different k · p theories at various high symmetry points. Group theoryplaces constraints – “compatibility relations” – on how this can be done. Each solution to these compatibility relationsgives groups of bands with different connectivities, corresponding to different physically-realizable phases of matter(trivial or topological). We solve all compatibility relations for all 230 space groups (SGs) by mapping connectivity inband theory to the graph-theoretic problem of constructing multipartite graphs. Classifying the allowed connectivitiesof energy bands becomes a combinatorial problem of graph enumeration: we present a fully tractable, algorithmicsolution.Second, we develop the tools to compute how the real-space orbitals in a material determine the symmetry characterof the electronic bands. Given only the Wyckoff positions and the orbital symmetry ( s, p, d ) of the elements/orbitalsin a material, we derive the symmetry character of all energy bands at all points in the BZ. We do this by extendingthe notion of a band representation (BR), first introduced in Refs. 2 and 3, to the physically relevant case of materialswith spin-orbit coupling (SOC) and/or TR symmetry. A BR consists of all bands linked to localized orbitals respectingthe crystal symmetry (and possibly TR). The set of BRs is strictly smaller than the set of groups of bands obtainedfrom our graph theory[4]. We identify a special subset of “elementary” BRs (EBRs)[2, 5], elaborated upon in theSupplementary Material (SM), which are the smallest sets of bands derived from local atomic-like Wannier functions.[6]We work out all the (10300) different EBRs for all the SGs, Wyckoff positions, and orbitals, which we will present ina separate data paper.[7]If the number of electrons is a fraction of the number of connected bands (connectivity) forming an EBR, thenthe system is a symmetry-enforced semimetal. The EBR method allows us to easily identify candidate semimetallicmaterials. As an amusing fact, we find that the largest possible number of connected bands in an EBR is 24 andhence the smallest possible fraction of filled bands in a semimetal is 1/24. If, however, our graph analysis reveals aninstance where the number of connected bands is smaller than the total number of bands in the EBR, we concludethat a k -space description exists but a Wannier one does not[8–10], i.e. the disconnected bands are topological.TIs are then those materials with bands that are not in our list of elementary components but are in our graphenumeration. We thus reformulate the momentum-space approach to topological indices as a real-space obstructionto the existence of atomic-like Wannier functions. In tandem with our graph-theoretic analysis of band connectivity,we enumerate all the ways the transition to a topological phase can occur. Hence, we are able to classify all topologicalcrystalline insulators. This leads to previously unrecognized large classes of TIs. We show the power of our approachby predicting hundreds of new TIs and semimetals. II. GRAPH THEORY AND BAND STRUCTURE
To construct a band theory which accounts for the global momentum space structure of energy bands, we piecetogether groups of bands from distinct points in the BZ. Consider a D -dimensional crystalline material invariantunder a SG G , containing elements of the form { R | d } , where R is a rotation or rotoinversion, and d is a translation.A Bravais lattice of translations is generated by a set of linearly independent translations { E | t i } , i = 1 . . . D , where E represents the identity. Each k vector in the reciprocal lattice is left invariant by its little group, G k ⊂ G , and theBloch wavefunctions | u n ( k ) i transform under a sum of irreps of G k ; bands at high-symmetry k -vectors will have (non-accidental) degeneracies equal to the dimension of these representations (reps). For spinless or spin-orbit free/coupledsystems, these are ordinary linear/double-valued reps. Away from high symmetry points, degeneracies are reducedand bands disperse according to conventional k · p theory.Consider two different high-symmetry k vectors, k and k , and a line k t = k + t ( k − k ), t ∈ [0 ,
1] connectingthem. To determine how the irreps at k connect to the irreps at k to form bands along this line, first note that thelittle group, G k t , at any k t on the line is a subgroup of both G k and G k . Thus, an irrep, ρ , of G k at k will split(subduce, or restrict) along the line to a (direct) sum of irreps L i τ i of G k t ; symbolically (here and throughout weuse “ ≈ ” to denote equivalence of representations) ρ ↓ G k t ≈ M i τ i . (1)These restrictions are referred to as compatibility relations . Heuristically, they are found by taking the representation ρ and “forgetting” about the symmetry elements of G k which are not in G k t . As energy bands in a crystal do notdiscontinuously end, the representations, σ , of G k at k must also satisfy σ ↓ G k t ≈ M i τ i . (2)Each representation, τ i , corresponds to a (group of) band(s) along the line k t ; bands coming from k in each irrep τ i must join with a group of bands transforming in the same irrep coming from k . We refer to each set of such pairingsas a solution to the compatibility relations.Compatibility relations apply to each and every connection line/plane between each pair of k points in the Brillouinzone, leading to strong but factorially redundant restrictions on how bands may connect in a crystal. To constructthe nonredundant solutions to the compatibility relations, we map the question to a problem in graph theory. Eachirrep at the different symmetry distinct k vectors labels a node in a graph. In our previous example, the nodes wouldbe labelled by ρ, σ, { τ , τ , . . . } for the k , k , and k t high symmetry points and line, respectively. We draw the edgesof the graph by the following rules: Irreps at the same k vector can never be connected by edges – our graph ismulti-partite. Nodes corresponding to irreps at k a and k b can be connected only if G k a ⊆ G k b or G k b ⊆ G k a (i.e. k -vectors are compatible). Edges must be consistent with the compatibility relations. For instance, Eq. (1)corresponds to an edge from the node labelled by ρ to each node labelled by τ i . We refer to such a graph as a connectivity graph .We developed an algorithm (described in the SM) that outputs all distinct connectivity graphs for all SGs– agargantuan task. The factorial complexity is handled by several subroutines, which ensure that the minimal setof paths in momentum space is considered. Additional filters remove redundant or isomorphic solutions to thecompatibility relations. The tools of graph theory then allow us to partition the nodes of the graph (the little-groupirreps) into distinct connected components (subgraphs). Each component corresponds to a connected, isolated groupof bands that can describe a set of valence bands in some insulating system or protected semimetal, depending onthe filling. In particular, such a list consists of all (both topologically trivial and nontrivial) valence band groups.The familiar example of graphene with SOC is given in Fig. 1 and the SM. We now define and classify topologicallynontrival bands in terms of localized
Wannier functions.
III. TOPOLOGICALLY (NON)TRIVIAL BANDS
Consider a group of connected bands in the spectrum of a crystal Hamiltonian separated by a gap from all others.Using existing machinery, to determine whether this group is topologically nontrivial requires discovering topologicalinvariants (indices or Wilson loops) from the analytic structure of the Bloch eigenfunctions. We now prove that the algebraic global structure of the energy spectrum itself (including connectivities) contains a complete classification oftopological materials. We define: • Definition 1.
An insulator (filled group of bands) is topologically nontrivial if it cannot be continued to any atomic limit without either closing a gap or breaking a symmetry.To every isolated group of energy bands, we associate a set of Wannier functions – orbitals obtained by Fouriertransforming linear combinations of the Bloch wavefunctions. In an atomic limit, the Wannier functions are expo-nentially localized, respect the symmetries of the crystal (and possibly TR) and coincide in most cases (however, seeSection V) with the exponentially localized atomic orbitals at infinite (atomic limit) separation. Under the action ofthe crystal symmetries, different atomic sites are distributed into orbits, belonging to Wyckoff Positions (WPs); wedenote the points in a Wyckoff orbit in a unit cell as { q i } . In analogy with the symmetry group of a k -vector, to eachsite q i there is a finite subgroup, G q i , of the full SG, G , which leaves q i invariant, called the site-symmetry group. Forexample, the A, B sites in graphene belong to WP 2 b (the multiplicity 2 refers to the number of symmetry-related sitesin the unit cell); its site-symmetry group is isomorphic to C v . Wannier functions at each site q i transform under somerep, ρ i , of G q i . Crucially, through the mathematical procedure of induction, the real-space transformation propertiesof these localized Wannier functions determine the little group reps of the bands at every point in the BZ: the actionof the SG on the full lattice (rather than just the unit cell) of Wannier functions gives an infinite-dimensional rep.Its Fourier transform gives the k dependent matrix rep of all symmetry elements. This restricts to reps of the littlegroup of each k -vector. Following Zak [2], we refer to this as a band representation (BR), ρ iG , induced[11] in thespace-group G by the rep ρ i of G q i : ρ iG = ρ i ↑ G. (3)The above is true with or without TR symmetry. BRs which also respect time-reversal symmetry in real space are physical band representations (PBRs).By inducing BRs, we enumerate all groups of bands with exponentially localized and symmetric Wannier functions.Each such group forms a BR, and every band representation is a sum of EBRs. We have identified the 10300 EBRs: • Proposition 1.
A band representation ρ G is elementary if and only if it can be induced from an irreduciblerepresentation ρ of a maximal (as a subgroup of the space group. See the SM) site-symmetry group G q whichdoes not appear in a list of exceptions[4, 12] given in the SM.We prove a similar statement for physically elementary band representations (PEBRs) and work out their extensivelist of exceptions.[7] In analogy with irreps, an (P)EBR cannot be written as a direct sum of other (P)BRs. A BRwhich is not elementary is a “composite” BR (CBR). Crucially, any topologically trivial group of bands is equivalentto some (P)BR, a conclusion of Definition 1. Conversely, any topologically nontrivial group of bands cannot beequivalent to any of the enumerated (P)BRs (a caveat is discussed in Section V).We thus conclude that the Wannier functions of a topologically trivial group of bands are smoothly continuable intoan atomic limit, exponentially localized, and transform under a BR. In a topological material, the Wannier functionsfor the valence (group of) bands either fail to be exponentially localizable, or break the crystal symmetry.An example of is the Chern insulator: a nonvanishing Chern number is an obstruction to exponentially localizedWannier functions.[13] The Kane-Mele model of graphene[14] (Section IV), is an example of : in the Z nontrivialphase, exponentially localized Wannier functions for the valence bands necessarily break TR symmetry[8] (when thevalence and conduction bands are taken together, atomic-like Wannier functions can be formed). Hence, in order totransition to the atomic limit a gap must close.We have tabulated all the (P)EBR’s induced from every maximal WP (i. e. a WP with maximal site-symmetrygroup) for all 230 SGs, with and without SOC and/or TR symmetry. This data allows us to enumerate all topologicallytrivial band structures. In an accompanying publication[7] we describe the myriad group-theoretic data (subductiontables, etc) for each of the EBRs and
PEBRs that we find . To generate this data, we generalized thewell-known induction algorithm based on Frobenius reciprocity and presented in Refs. 5 and 15 to the case of double-valued representations. We present the full details of the computational methods in Ref. 7. Additionally, the datacan be accessed through programs hosted on the Bilbao Crystallographic Server[16]. We also give a summary tableof all EBRs and PEBRs in Sec. VII of the Supplementary MaterialThis allows us to give, for the first time, a classification of TIs that is both descriptive and predictive . Rather thanproviding a classification (with a topological index, for instance) of the topological phases in each space group, divorcedfrom predictive power, we instead formulate a procedure to determine whether a band structure is topologically trivialor topologically nontrivial, and enumerate the possible non-trivial band structures for each space group. Given theband structure of a material, we can compare isolated energy bands to our tabulated list of (P)EBRs. Any groupof bands that transforms as a (P)EBR is topologically trivial. Those groups of bands that remain are guaranteed tobe topologically nontrivial. Conversely, knowing just the valence orbitals and crystal structure of a material, we canimmediately determine under which – if any – EBRs the bands near the Fermi level transform. If graph theory revealsthat these EBRs can be disconnected, we deduce that this phase is topological. While a disconnected EBR itself servesas a topological index, standard techniques can be used to diagnose which (if any) of the more standard K-theoretictopological indices[17] are nontrivial. In the subsequent sections, we outline this recipe, and present hundreds of newpredicted topological materials.Our identification of topological crystalline phases also goes beyond recently proposed classifications based onsymmetry eigenvalues of occupied bands in momentum space, first proposed in Ref. 18, and applied in a modifiedform to three dimensional systems in Ref. 19. A shortcoming common to both methods is that while they produce alist of topological indices for each space group, they provide no insight into how to find or engineer materials in anynontrivial class. Second, in constrast to the claims of Ref. 19, by focusing only on eigenvalues in momentum spacerather than the real-space structure of Wannier functions (or equivalently, the analytic structure of Bloch functionsin momentum space), essential information about the topological properties of certain band structures is lost. Forinstance, the topological phases of SG P mm (183) presented in the Supplementary Material fall outside the scope ofRef. 19. Viewed in this light, our classification based on band representations generalizes the notion of a topologicaleigenvalue invariant in such a way as to capture these missing cases. IV. CLASSIFICATION OF TIS
Combining the notion of a BR with the connectivity graphs, we identify several broad classes of TI’s, distinguishedby the number of relevant EBRs at the Fermi level when transitioning from a topologically trivial to a nontrivialphase. If, in a trivial phase, the Fermi level sits in a single (P)EBR, the material is necessarily a semimetal. If tuningan external control parameter (strain, SOC, etc), opens a gap at the Fermi level, the material necessarily becomestopological. This is because if an (P)EBR splits into a disconnected valence and conduction band, then neither thevalence nor the conduction band can form BRs: only both together form a (P)EBR. This situation occurs exactlywhen an EBR can be consistently realized in a disconnected way in the BZ. This is precisely the sort of task suitedfor the graph-theoretic machinery of Section II! The archetypal example for this behavior is the quantum spin Halltransition in the Kane-Mele model of graphene with both next nearest neighbor “Haldane”[20] and Rashba SOC,which we illustrate schematically in Fig. 1. This model hosts two spinful p -orbitals per hexagonal unit cell, for a totalof four bands forming an EBR. In the Rashba SOC regime, all four bands are connected and the material is a gaplesssemimetal. Turning up Haldane SOC opens a band gap. By the preceding analysis, this gap must be topological.In Ref. 7, we give all (hundreds of) cases – each specified by an orbital, Wyckoff position, and SG – where this canoccur. We give the full details in the SMThe second class of TI’s is defined by the presence of more than one relevant EBR at the Fermi level. The trivialphase of such a material can be an insulator, with EBRs above and below the Fermi level. Without loss of generality,we consider one EBR in the conduction band and one in the valence band; generically, any transition involvingmore than two EBRs can be resolved into a sequence of pairwise transitions. A topological phase transition occurswhen a gap closes and reopens after a band inversion, such that neither the valence bands nor the conduction bandsform a BR. In the trivial phase, the little group irreps of the filled bands at each k point are those of the valenceband EBR. After the topological phase transition, the little group irreps at each k point are not consistent with anEBR. This mechanism describes the zeitgeist 3 D TI Bi Se .[21, 22] Without, or with very small, SOC, Bi Se is atrivial insulator; its valence and conduction bands transform in two distinct EBRs. Increasing SOC pushes the bandstogether; at a critical strength, the gap between the valence and conduction bands closes at the Γ point of the BZ.Above this critical value, a gap reopens, with certain states (labelled by irreps of G Γ ) exchanged between the valenceand conduction bands. Ultimately, neither the valence bands nor the conduction bands transform as EBRs and theinsulator is topological as per Def. 1[23]. If, on the other hand, a full gap does not reopen after band inversion, thenwe are left with symmetry-protected semimetal a la Cd As [24] or Na Bi[25].When the phase transition is driven by SOC, we label the classes by ( n, m ), where n is the number of EBRs at theFermi level in the trivial phase (without SOC) and m is the number in the topological phases (after SOC is turnedon). These phases are indicated in Table I, along with material examples.To summarize the theoretical results: in Section II we showed that the constraints placed by group theory in momentum space on the allowed connectivity of bands can be solved via a mapping to graphs. We constructedall possible allowed isolated band groupings for all 230 SGs. We then showed that our group-theoretic analysis in real space determines – through the notion of BRs – which of these isolated band groups are described by localizedsymmetric Wannier functions (topologically trivial insulators.) It follows that any other group of valence bandsin an insulator necessarily constistute a TI. Importantly, our theory also shows that there exist different classes oftopologically trivial insulators, which cannot be adiabatically continued to one another. We now link this importantfact to orbital hybridization. V. CHEMICAL BONDING, HYBRIDIZATION, AND NON-EQUIVALENT ATOMIC LIMITS
Given a topologically trivial crystal, it is tempting – but wrong – to assume that the electronic Wannier functions,like the constituent atomic orbitals, are localized at the atomic positions. Basic chemistry informs us that orbitalsfrom different atomic sites can hybridize to form bonding and antibonding “molecular” orbitals, centered away fromany individual atom[26]. In a crystal formed of these tightly bound molecular (rather than atomic) units, the valenceand conduction BRs are induced from the (generically maximal) WPs of the molecular orbitals, rather than fromthe atomic orbitals; consequently, the valence and conduction band Wannier functions are localized at the molecularorbital WPs, away from the atoms.Thus, orbital hybridization, when viewed in the solid-state, represents the required transition between two symmetrydistinct atomic limit phases. In both phases localized, symmetric Wannier functions exist; the distinction lies in wherethe orbitals sit in the atomic limit[27, 28]. In the first atomic limit, the orbitals lie on the atomic sites. In the second,however, the orbitals do not coincide with the atoms. This phase has been called topological[29, 30], but we referto it as an “obstructed” atomic limit, since it is also described by localized Wannier states[10, 31]. The prototypicalexample is the 1 D Su-Schrieffer-Heeger (Rice-Mele) chain, whose two phases are distinguished by their hybridizationpattern, an electric dipole moment that is 0 ( ) in the trivial (nontrivial) phase. Pumping between two differentatomic limits always leads to a nontrivial cycle with observable transport quantities.[32–34] A similar phenomenon isobserved in the newly discovered quadrupole insulators.[35] Signatures of the obstructed atomic limit also appear inthe real-space entanglement structure of the insulating ground state in finite systems[36], and these signatures persistto systems containing only a single molecule.[37]We now describe orbital hybridization with EBRs. The valence and conduction bands are each described by anEBR. Taken together, the two EBR’s comprise a CBR. In both the first and the second atomic limit, as well as at thecritical point between them, the CBR does not change, although the individual EBR’s of the valence and conductionbands do. We denote the CBR in the first atomic limit as σ v ↑ G ⊕ σ c ↑ G , where σ v and σ c are irreps of the sitesymmetry group, G a , of the atomic sites. In the second atomic limit, the CBR is ρ v ↑ G ⊕ ρ c ↑ G , where ρ v and ρ c are irreps of the site symmetry group, G m , of the molecular sites. As we show in the SM, a symmetry-preservingtransition can only happen when the two site symmetry groups, G a and G m , have a common subgroup G with arepresentation η such that η ↑ G a ≈ σ v ⊕ σ c , η ↑ G m ≈ ρ v ⊕ ρ m . (4)This equation indicates that there is a line joining the atomic sites, and that Wannier functions localized along theline give the same set of bands as those localized at either endpoint. Because of this, the Wannier functions for thevalence band can move from the atomic to the molecular sites while preserving all symmetries upon passing througha critical point. In the SM, we illustrate this using the example of sp orbital hybridization. Note, furthermore, thatwe can define an analogous notion of an obstructed atomic limit for systems lacking translational symmetry, usingproperties of the point group only. In this way, the preceding discussion generalizes straightforwardly to finite-sizedcrystals, molecules, and even quasicrystals. In the latter case, obstructions to the naive atomic limit are preciselycovalent bonds. We conclude that hybridization, and thus chemical bonding, can be treated as a phase transition. VI. ALGORITHMIC MATERIALS SEARCH
We demonstrate the power of our theory by proposing two algorithms that use databases – such as the InorganicCrystal Structure Database (ICSD)[38] – which tabulate the occupied WPs for each element in a chemical compound,along with simple energy estimates, in order to identify many new classes of TIs and semimetals. We distinguishbetween the number of EBRs at the Fermi level, i.e., the two classes defined in Section IV and summarized in Table I.We further distinguish between cases with and without SOC, which can be treated separately by existing ab-initiomethods. The SOC strength can be viewed as a control parameter driving the topological phase transition.For EBRswithout SOC, we count bands on a per-spin basis: all states are doubly degenerate. Further new topologicalsemimetals, such as a series of ones at filling -1/8, can be obtained by our method.
A. Single PEBRs at the Fermi Level
As described in Section IV and Table I, a (1 ,
1) type TI occurs when a single PEBR is realized as a sum of twoband groups disconnected in momentum space. Here, we utilize the fact that (1 ,
1) TIs can be realized by bandrepresentations induced from 1 D site symmetry group irreps that are EBRs but not PEBRs. In this case, theWannier functions for the valence band involve a single orbital per site, and hence do not respect the twofold time-reversal symmetry degeneracy in real space : since their Wannier states break time-reversal symmetry the material isnecessarily a TI.An example is furnished by lead suboxide Pb O[39], a material whose topological properties were until now unex-plored. As discussed in the Supplementary Material, this is a non-symmorphic cubic crystal in the space group
P n ¯3 m (224). Although metallic, this material features a topologically disconnected PEBR far below the Fermi level, shownin Fig. 2 a . However, we can consider the application of z − axis uniaxial strain, under which the crystal symmetry islowered to the tetragonal subgroup P /nnm (134) of the original space group. There are fewer symmetry constraintsimposed on the band structure in this space group, and in particular degeneracies protected by threefold rotationalsymmetry will be broken. This allows for a gap to open at the Fermi level, leading us to predict that strained Pb Owill be a topological insulator with a small Fermi pocket, as shown in Fig. 2 b . As an aside, we note that this analysisshows that by using group-subgroup relations, we can make predictions about the topological character of strainedsystems when the unstrained band structure is known.Cu SbS , a candidate TI[40, 41], is also an example of a (1 ,
1) type material. In Fig. 2 c we show the calculated bandstructure. The states near the Fermi level form a single EBR coming from the Cu d -orbital electrons. With SOC, theEBR is gapped, leading to topologically nontrivial valence and conduction bands. A trivial spurious EBR also liesenergetically within the topological gap (shown in black in the inset to Fig. 2 c ), causing the effective transport gapto be smaller than the topological gap (similar to HgTe). There are 35 additional materials in this Cu ABX class ofmaterials.We use these considerations to design a systematic method to search for (1 ,
1) type TI’s. First, we identify alldisconnected PEBRs from our data paper Ref. 42, which is organized by WP and site-symmetry irrep; in the SMwe indicate which orbital types give rise to these representations. Next, we cross-reference with the ICSD[38], whichyields a list of candidate materials. This list can be further reduced by restricting to semi-metals (since an insulatorwould not yield a topological gap near the Fermi level after turning on SOC). Finally, an electron counting analysis,using only the atomic-limit orbital energies, will determine whether or not the topologically relevant BRs will lie nearthe Fermi level. Carrying out this procedure led us to identify the Cu ABX material class introduced in the previousparagraph. We also provide a list[43] of materials guaranteed to be semi-metals by the minimal-insulating-fillingcriteria[44] (a sufficient, but far from necessary condition for a semi-metal). B. Multiple PEBRs
As described in Section IV, a CBR can be disconnected in such a way that neither the valence nor the conductionbands form PEBRs. When these are double-valued (spinful) BRs, we classify them by whether the BR is compositeor elementary without SOC. In the first case, a single, connected, SOC-free EBR decomposes after turning on SOCinto a sum of two PEBRs that are disconnected in a topologically nontrivial way. These constitute (1 ,
2) type TIs.This class includes materials composed of layered Bi − square nets and related structures, which we discuss further inthe SM. The relevant states at the Fermi level are the Bi p x and p y orbitals, which induce a single EBR when SOC isneglected. These materials are filling-enforced semimetals with a symmetry-protected nodal (degenerate) line at theFermi level. SOC fully gaps the nodal line, causing the EBR to disconnect. Consequently, this is a topological gap.Fig. 3 depicts band structures for the representative TIs SrZnSb and ZrSnTe[45, 46].A similar materials search program to that described in the previous subsection can be implemented for the (1 , ,
2) materials in SG P /nmm (129)[45–47], as well as 58 new candidate TIs in SG Pnma (62). SG
Pnma (62)arises from an in-plane distortion of SG P /nmm (129), and again demonstrates that our group-theoretic approachallows us to predict the topological character of materials upon structural distortion. We present these materials inthe SM.Finally, we discuss the known TIs Bi Se and KHgSb. In these materials, without SOC the band representationat the Fermi level is gapped and composite. With infinitesimal SOC, the band representation remains gapped andcomposite. A topological phase transition occurs only when SOC is strong enough to drive a band inversion thatexchanges states with distinct little group reps at Γ between the conduction and valence bands. Because such atransition depends on the strength of SOC – unlike in the (1 ,
1) and (1 ,
2) cases, where the transition is at infinitesimalSOC – Bi Se and KHgSb are described by our method but could not be unequivocally predicted. C. Partially filled bands and semimetals
Although not the main focus of this work, we also note that our method allows for the prediction of metals andsemimetals. Going beyond standard methods of counting multiplicities of occupied WPs[26], we can predict symmetry-protected semimetals by looking for partially-filled, connected EBRs induced from high-dimensional site-symmetryrepresentations. In this way we have found the A B family[48] of sixteen-fold connected metals in SG I ¯43 d (220)with A=Cu,Li,Na and B=Si,Ge,Sn,Pb. Through charge-transfer, the sixteen bands in this PEBR are 7 / Ge is shown in Fig. 2 d . There is a symmetry-protected threefold degeneracy near the Fermi level at the P point[49]Finally, we calculate that Cu TeO in SG Ia ¯3 (206)[50] has a half filled, connected twenty-four band PEBR at theFermi level when interactions are neglected. Our EBR theory reveals that this is the highest symmetry-enforced bandconnectivity in any SG. D. Systematic materials search: Summary
While the materials presented above represent the proof-of-principle for our material search strategy, a full, sys-tematic search of the entire materials database based on our criteria reveals a myriad of new topological insulatorand semimetal candidates. While we defer a full discussion of our new materials predictions to a forthcoming work,we shall here list some of our more promising candidate materials, as identified through our systematic search, andverified with ab-intio DFT calculations. In SG P¯3 m Ge , CeISi, BiTe, and Nb Cl willbe topological insulators. In SG P /mmm (123) we have LiBiS and AgSbTe . A particularly promising familyof topological materials is given by TiAsTe, ZrSbTe, HfSbTe, Hf Ni Ge , Sr Li Sb , Ba Bi , and Ba Al Ge in SG I/mmm (71). Additionally, we find NaAu in SG F d ¯3 m (227), LaPd O in SG I /a (88), BaGe Ru in SG F ddd (70), Ni Ta Te in SG P bam (53), Ag Ca Si in SG F mmm (69), Ta Se I in SG I
422 (97), SnP in SG I mm (107),and Tl CuTe in SG I /mcm (140) each of which we predict hosts novel topological bands. VII. CONCLUSION
We have combined group theory, chemistry, and graph theory to provide the framework of Topological QuantumChemistry. We provide a complete description of the Wannier-Bloch duality between real and momentum space,in the process linking the extended (physics) versus local (chemistry) approaches to electronic states. Our theoryis descriptive and predictive: we can algorithmically search for and predict new TIs and semimetals. In a series ofaccompanying Data papers, we present the group-theoretic data and graph-theoretic algorithms necessary to deducethe conclusions of Sections II and III, and to implement the materials search described in Section VI. By taking theideas presented in this paper to their logical conclusion, we arrive at a new paradigm, which applies not only to TIs,but to semimetals and to band theory in general. The synthesis of symmetry and topology, of localized orbitals andBloch wavefunctions, allows for a full understanding of noninteracting solids, which we have only begun to explore inthis work. Furthermore, our emphasis on the symmetry of localized orbitals opens a promising avenue to incorporatemagnetic groups or interactions into the theory of topological materials.
Data Availability:
All data supporting the conclusions of this work is hosted on the Bilbao CrystallographicServer ( http://cryst.ehu.es ). All information about EBRs, PEBRs, and their connectivity graphs can be accessedvia the BANDREP application[16]. The algorithms used to generate this data, as well as a guide to the use of allrelevant programs, can be found in the accompanying data papers, Refs. 7 and 42.
Acknowledgements:
BB would like to thank Ivo Souza, Richard Martin, and Ida Momennejad for fruitful discus-sions. MGV would like to thank Gonzalo Lopez-Garmendia for help with computational work. BB, JC, ZW, and BABacknowledge the hospitality of the Donostia International Physics Center, where parts of this work were carried out.JC also acknowledges the hospitality of the Kavli Institute for Theoretical Physics, and BAB also acknowledges thehospitality and support of the ´Ecole Normale Sup´erieure and Laboratoire de Physique Th´eorique et Hautes Energies.The work of MVG was supported by FIS2016-75862-P and FIS2013-48286-C2-1-P national projects of the SpanishMINECO. The work of LE and MIA was supported by the Government of the Basque Country (project IT779-13)and the Spanish Ministry of Economy and Competitiveness and FEDER funds (project MAT2015-66441-P). ZW andBAB, as well as part of the development of the initial theory and further ab-initio work, were supported by theDepartment of Energy de-sc0016239, Simons Investigator Award, the Packard Foundation, and the Schmidt Fund forInnovative Research. The development of the practical part of the theory, tables, some of the code development, andab-initio work was funded by NSF EAGER Grant No. DMR-1643312, ONR - N00014-14-1-0330, and NSF-MRSECDMR-1420541.
Author Contributions:
BB, LE, JC, MGV, and ZW contributed equally to this work. BB, JC, ZW and BABprovided the theoretical analysis, with input from CF. JC developed specific models to test the theory. LE andMIA performed the computerized group-theoretic computations. BB, LE and MGV devised and developed the graphalgorithms, as well as the EBR connectivities; LE and MGV performed the computerized graph theory computations.ZW discovered the new materials presented in this paper with input from CF, and performed all first-principlescalculations.
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Disconnected EBRWannier FunctionsConnected EBR k·p theoryatomic orbitals Graph Theory SubductionInduction (BR)Topological semi-metal Topological insulator
FIG. 1.
Schematic: how our theory applies to graphene with SOC . We begin by inputting the orbitals ( | p z ↑i , | p z ↓i )and lattice positions relevant near the Fermi level. Following the first arrow, we then induce an EBR from these orbitals, whichsubduces to little group representations at the high symmetry Γ , M and K points, shown here as nodes in a graph. Standard k · p theory allows us to deduce the symmetry and degeneracy of energy bands in a small neighborhood near these points - thedifferent colored edges emanating from these nodes. The graph theory mapping allows us to solve the compatibility relationsalong these lines in two topologically distinct ways. On the left, we obtain a graph with one connected component, indicatingthat in this phase graphene is a symmetry-protected semimetal; the Wannier functions for the four connected bands coincidewith the atomic orbial Wannier functions. In contrast, the graph on the right has two disconnected components, correspondingto the topological phase of graphene by Def. 1. The spin up and spin down localized Wannier functions for the valence bandare localized on distinct sites of hexagonal lattice, and so break TR-symmetry in real space[8]. a X M R X|M R -4-3-2-1012 E n e r gy ( e V ) soc E F b X M R X|M R -4-3-2-1012 E n e r g y ( e V ) soc_pz E F c X M Z A R Z -2-1.5-1-0.500.51
Cu3SbS4 E F -0.2-0.100.10.2 E n e r gy ( e V ) zoom E F ( ) d H N P H -3-2-1012 E n e r g y ( e V ) soc E F Γ Γ Γ Γ Γ Γ Γ Γ FIG. 2.
Representative band structures for new material predictions . a Band structure of Pb O in SG
P n ¯3 m (224).The red group of bands 3 eV below the Fermi level originate from a topologically disconnected PEBR. b Band structure forPb O under uniaxial strain along the z -axis. This distortion opens a topological gap near the Fermi level. c Band structurefor the topologically nontrivial compound Cu SbS . The conduction band, induced from a 1 D site-symmetry irrep, is not aPEBR. The inset shows a zoomed in view of the gap at Γ. d Band structure for the twenty-fourfold connected (semi-)metalCu TeO in SG Ia ¯3 (206). The bands at the Fermi level form a 24-band PEBR which is half-filled. a X S Y -3-2-101 E n e r g y ( e V ) ZrSnSb2 E F b X M Z -3-2-101 E n e r g y ( e V ) ZrSnTe E F Z Γ Γ Γ Γ FIG. 3.
Representative band structures for topologically nontrivial insulators in the Bi- square net structuretype. a shows the band structure of the 3 D weak TI SrZnSb in SG Pnma (62), where there is a small in-plane distortion inthe Sb square net. b shows the band structure of the 3 D weak TI ZrSnTe in SG P4/nmm (129). upplementary Material for Topological Quantum Chemistry
I. BAND REPRESENTATIONS AND WANNIER FUNCTIONS
Here we expand upon the notion of band representations as discussed in the main text. First introduced by Zak , aband representation for a space group G is a set of energy bands E n ( k ) spanned by a given collection of (exponentially)localized Wannier orbitals. To be consistent with the crystal symmetries, the localization centers of these Wannierorbitals (Wannier centers) must form an orbit under the action of G : given a Wannier function centered at a point q in the unit cell, there is a Wannier function centered at g q for all g ∈ G , including all positions related to these bylattice translations. The set { q α } of these Wannier centers form an orbit under the space group action and hence canbe labelled by a Wyckoff position of the space group G . Note that every system representable with a tight-bindingmodel has such a real-space [direct space] description.Before we show how to construct a band representation from a set of localization centers of the Wannier orbitals,we will carefully define our terminology. Note that we use the conventional origin choice (origin choice 2) for all spacegroups as given by the Bilbao Crystallographic Server . Our terminology follows that of Refs. 2–7. For basic factsabout the theory of finite groups, we refer the reader to Refs. 8 and 9. Definition 1. A symmetry site q is any point in the unit cell of a crystal. The set of symmetry operations g ∈ G that leave q fixed (absolutely, not up to lattice translations) is called the stabilizer group , or site-symmetry group G q ⊂ G . By definition, a site-symmetry group is isomorphic to a crystallographic point group. A site-symmetry groupis called non-maximal if there exists a finite group H , such that G q ⊂ H ⊂ G ; a site-symmetry group that is notnon-maximal is maximal . Note that the translation part of g ∈ G q may include lattice translations, so long as it keeps the point q fixed.Nonetheless, G q must be isomorphic to a point group. Definition 2.
The orbit { q α = g α q | g α / ∈ G q } , α = 1 , . . . , n of a symmetry site q modulo lattice translations areclassified by a Wyckoff position of multiplicity n . Note that we define the multiplicity with respect to the primitive,rather than the conventional cell. The stabilizer groups G q α are all isomorphic and conjugate to the stabilizer group G q ≡ G q . We say that a Wyckoff position is maximal if the stabilizer group G q is maximal. As an example, if q is a general point in the unit cell with trivial stabilizer group, G q = { E | } , then q belongsto the “general” Wyckoff position with multiplicity equal to the order of the point group of the space group. This isnot a maximal position in general, but it is a position.Let us now return to the problem of constructing a band representation. Without loss of generality, consider thecase of Wannier functions localized on symmetry sites { q α | α = 1 , ..., n } classified by a single Wyckoff position ofmultiplicity n . Then the n q functions localized on the site q ≡ q transform under some representation ρ of thesite-symmetry group G q , with dimension n q . For the time being we do not specify whether or not ρ is irreducible; wewill show later that we need only concern ourselves with the irreducible representations (irreps). Crystal symmetrydictates that there are n q orbitals localized on the other equivalent symmetry sites q α in the orbit, and that thesetransform under the conjugate representation defined by ρ α ( h ) = ρ ( g − α hg α ) (S1)for h ∈ G q α . One can see that g − α hg α ∈ G q because h ∈ G q α ⇒ hq α = q α ⇒ hg α q = g α q . We can thus index ourWannier functions as W iα ( r − t µ ), where i = 1 . . . n q indexes the functions localized on symmetry site q α + t µ and t µ is a lattice vector.Finally, the elements g α , α = 1 act by permuting the different symmetry sites q α . Taking all of these facts togetherallows us to define the induced representation ρ G ≡ ρ ↑ G ≡ Ind GG q ρ of the space group G induced from therepresentation ρ of G q . This representation is n q × n × N dimensional (assuming periodic boundary conditions),where N → ∞ is the number of (primitive) unit cells in the system, and the representation matrices have a blockstructure, with n × N blocks of n q × n q dimensional submatrices; a group element g whose matrix representativehas a nonvanishing ( αβ ) block maps q β to q α .For our purposes, it is most convenient to work with the Fourier transforms a iα ( k , r ) = X µ e i k · t µ W iα ( r − t µ ) , (S2) a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l where the sum is over the lattice vectors and α = 1 , ..., n . In this way we can exchange our infinite N × n × n q matrices for finite-dimensional n × n q matrix-valued functions of k , which takes N values in the first Brillouin zone(BZ). Any translationally-invariant, quadratic Hamiltonian acting in the Hilbert space of these Wannier functionscommutes with these matrices. The concrete formula for the induced representation matrices ρ G ( g ) can then bedefined as follows: Definition 3.
The band representation ρ G induced from the n q − dimensional representation ρ of the site-symmetrygroup G q of a particular point q , whose orbit belongs to the Wyckoff position { q α ≡ g α q | g α / ∈ G q for α = 1 } ofmultiplicity n , is defined for all h ∈ G by the action ( ρ G ( h ) a ) iα ( k , r ) = e − i ( h k ) · t βα n q X i =1 ρ i i ( g − β { E | − t βα } hg α ) a i β ( h k , r ) , (S3) here α, β, i, j are matrix indices, where for each choice of α the index β is determined by the unique coset of G thatcontains hg α : hg α = { E | t βα } g β g (S4) for some g ∈ G q and Bravais lattice vector t βα . By moving g α to the right-hand-side of Eq (S4), it is evident that h q α = { E | t βα } g β gg − α q α = { E | t βα } g β g q = { E | t βα } g β q = { E | t βα } q β . The second and fourth equalities follow fromthe definition of q α,β and the third equality follows from g ∈ G q . Thus, t βα = h q α − q β . (S5) If ρ is an n q -dimensional representation of G q , and if the Wyckoff multiplicity of the position { q α } is n , then thereare n × n q energy bands in the band representation ρ G . (This is a special case of the general induction procedure; a similar formula can be used for inducing the repre-sentation of any group from one of its subgroups.) Notice that we did not need to specify a particular Hamiltonian.Our discussion applies to any Hamiltonian that respects the crystal symmetry and acts on a local Hilbert space ofWannier functions. All of the above holds for either spinless or spin-orbit coupled systems. When spin-orbit couplingis negligible, we consider the single valued (or spinless) linear representations of the site symmetry group; we doubleeverything to account for the trivial spin degeneracy. For spin-orbit coupled systems, we must use the double-valued(spinor) representations.Note that a band representation is formally infinite dimensional since it depends on the momentum k (it has asmany dimensions as the number of unit cells in the crystal), while the irreducible representations of the space groupsare indexed by discrete sets of k vectors. As such, every band representation is formally reducible, as it decomposesas an infinite direct sum (over k points) of space group irreps at each k point. However, it is the band representations,rather than the space group irreps, that tell us about the global band structure topology in a crystal. Hence, we willbe interested in the decomposition of band representations into sums of other band representations.First, we must specify how to tell if two band representations are equivalent. Given two band representations ρ G and σ G , a necessary condition for their equivalence up to now is that at all points in the BZ they restrict to the samelittle group representations. However, for the study of topological phases, we need a stronger form of equivalence. Wedefine a form of homotopy equivalence that makes explicit the smoothness properties needed for discussing topologicalphase transitions. Namely, we say Definition 4.
Two band representations ρ k G and σ k G are equivalent iff there exists a unitary matrix-valued function S ( k , t, g ) smooth in k and continuous in t such that for all g ∈ G S ( k , t, g ) is a band representation for all t ∈ [0 , ,2. S ( k , , g ) = ρ k G ( g ) , and3. S ( k , , g ) = σ k G ( g )Note that since S is continuous in t , any property of a band representation evolves continuously under the equivalence S . In particular, the Wilson loop (i. e. Berry phase) matrices computed from the bands in the representation ρ k G are homotopic to the Wilson loop matrices computed in the representation σ k G . As such, two equivalent bandrepresentations cannot be distinguished by any quantized Wilson loop invariant. This is a constructive formulationof the type of equivalence noted in Refs. 14 and 10We can also give a more constructive view of equivalence. Consider two symmetry sites, q , q , which have distinct sitesymmetry groups, G q and G q , respectively, with nonempty intersection, G = G q ∩ G q . Since G is the intersectionof two distinct stabilizer groups, it is a stabilizer group of some (lower) symmetry site q . This symmetry site willhave a variable parameter that interpolates between the symmetry sites q and q ; this allows us to easily identify q from a table of Wyckoff positions, which we have done for all the Wyckoff positions of all the 230 space groups. If G is an index m q subgroup in G q , then the associated Wyckoff position with stabilizer group G has multiplicity m q times that of q . Furthermore, G q has a coset decomposition in terms of m q cosets of G ; analogous statements holdwhen we view G as an index m q subgroup of G q . We can use this coset decomposition to induce representations of G q and G q from representations of G , in much the same way as outlined above. Given a representation ρ of G , theband representations ( ρ ↑ G q ) ↑ G and ( ρ ↑ G q ) ↑ G are equivalent. The existence of a homotopy S implementingthis equivalence is guaranteed by the fact that the symmetry site q can be continuously moved from q to q withoutviolating the crystal symmetries.Using Definition 4 of equivalence, we define Definition 5.
A band representation is called composite if it is equivalent to the direct sum of other band represen-tations. A band representation that is not composite is called elementary . Using the fact that induction of representations, ↑ , commutes with direct sums, and that induction factors throughsubgroup inclusion , we deduce that elementary band representations are induced from irreducible representationsof maximal site symmetry groups. These conditions are necessary, however they are not sufficient. It may still be thecase that a band representation induced from a maximal site symmetry irrep is equivalent [in the sense of Def. (4)]to a composite band representation. We catalogue all such exceptions in Table S10 for the space groups (first foundin Ref. 5), and in Table S11 for the double space groups. The full decription of how this data was obtained will bepresented in the accompanying Data paper , and the data itself is accessible through the BANDREP program onthe Bilbao Crystallographic Server . We have Proposition 1.
A band representation ρ G is elementary if and only if it can be induced from an irreducible repre-sentation ρ of a maximal site-symmetry group G q which does not appear in the first column of Table S10 or S11. Up to this point, we have not commented on time-reversal symmetry. We may include antiunitary time-reversalsymmetry as an element in any site-symmetry group, as it acts locally in real [direct] space, i. e. it commutes with allspace group elements. For spinless systems, time-reversal squares to +1, while for spinful systems it squares to −
1. Wecall site-symmetry representations which are compatible with the action of time-reversal physical representations. Notethat all physically irreducible site-symmetry representations are even-dimensional for spinful systems by Kramers’stheorem. The entire discussion thusfar holds mutatis mutandis for physical band representations, physical equivalence,and physically elementary band representations, by generalizing Defs. 3,4, and 5 to the TR-symmetric case. Takingtime-reversal symmetry into account, we find that only the band representations below the double line in Table S10 failto be physically elementary. The rest of the entries in Table S10, as well as all the entries in Table S11 are compositewithout TR symmetry, but physically elementary. Moreover, we find there are additional physical exceptions forspinless systems with time-reversal symmetry, catalogued in Table S12. The machinery we used to obtain these tableswill be explained in detail in Ref. 16; they represent an exhaustive search of all induced representations for all spacegroups. Summarizing, we have with TR that
Proposition 2.
A spinless (i.e. single-valued) band representation ρ G is physically elementary if and only if it canbe induced from a physically irreducible representation ρ of a maximal site-symmetry group G q which does not appearin the first column of Table S10 below the double line, and if it does not appear in Table S12.A spinful (i.e. double-valued) band representation ρ G is physically elementary if and only if it can be induced froma physically irreducible representation ρ of a maximal site-symmetry group G q . II. CONNECTIVITY GRAPHS
In this Appendix, we review the necessary background for constructing the connectivity graphs associated withelementary band representations. After reviewing compatibility relations in more detail than presented in the maintext, we outline our algorithm for computing the allowed connectivities of elementary band representations usingthe notion of spectral graph partitioning. This allows us to develop an approach to the classification and search forTIs significantly more general than those found in recent proposals . A more complete account of the machineryand related data – which takes more than 100 pages – will be given in Ref. 20, but the following represents a goodintroduction to our method and its results.
A. Compatibility Relations
Recall that in the textbook theory of energy bands , global band topology – the various interconnections betweendifferent bands throughout the BZ – is inferred from the representations of the little groups G k through the use ofso-called “compatibility relations”. Specifically, irreducible little group representations at high-symmetry k -points,lines, and planes in the BZ are reducible along high-symmetry lines, planes, and volumes (the general k -point)respectively; the compatibility relations determine which representations can be consistently connected along thesesubspaces. Since the little groups along high-symmetry surfaces are subgroups of the little groups of their boundaries,the compatibility relations can be determined by starting with the little group representation on a high-symmetrysurface, and restricting (subducing) to the representations of the higher dimensional surfaces that it bounds.As an example, let us examine space group P ¯43 m (215). This is a symmorphic space group with primitive cubicBravais lattice, and point group T d . The group T d is generated by a threefold rotation, C , , a fourfold roto-inversion, IC z ≡ S − , and a mirror reflection, m . Consider the high-symmetry point Γ = (0 , ,
0) in the BZ, and the lineΛ = ( k, k, k ) emanating from it. The point group of the little group G Γ of Γ, known as the little co-group ¯ G Γ , isisomorphic to the point group of the space group, while the little co-group ¯ G Λ of Λ is generated by C , and m ,and thus isomorphic to the group C v . As such, irreps of G Γ restrict (or subduce) to representations of the littlegroup G Λ of the line Λ. The compatibility relations enumerate all such restrictions. For example, let us consider firstthe little group G Γ . We note that since the space group P ¯43 m (215) is symmorphic, the representations of the littlegroups G k are trivially determined by the representations of the little co-groups ¯ G k . Here and throughout, we willsimplify notation by giving representation matrices and character tables for the little co-groups where appropriate.The four dimensional double-valued ¯Γ representation of ¯ G Γ ≈ T d (this is the spin-3 / ( C , ) = √ e − iπ/ − i −√ i √ − i √ − − i −√ − i √ − i √ − i √ i √ − , ¯Γ ( IC z ) = − √− − ( − / √− − / ¯Γ ( m ) = − / − ( − / − ( − / − / (S6)The matrix for C , has eigenvalues − , − , e iπ/ , e − iπ/ , while the matrix for m has eigenvalues − i, − i, i, i .Next, we note that there are three double-valued representations of G Λ , conventionally labelled ¯Λ , ¯Λ , and ¯Λ .The matrix representative of { C , | } in each of these representations is given by¯Λ ( { C , | } ) = ¯Λ ( { C , | } ) = − , ¯Λ ( { C , | } ) = (cid:18) e − iπ/ e iπ/ (cid:19) , (S7)and the matrices for { m | } are¯Λ ( { m | } ) = − i, ¯Λ ( { m | } ) = i, ¯Λ ( { m | } ) = (cid:18) −
11 0 (cid:19) (S8)By comparing eigenvalues, we deduce that the ¯Γ representation of G Γ must restrict to the ¯Λ ⊕ ¯Λ ⊕ ¯Λ represen-tation of G Λ . The compatibility relation for the ¯Γ representation at Γ → Λ is thus,¯Γ ↓ G Λ ≈ ¯Λ ⊕ ¯Λ ⊕ ¯Λ (S9) B. Graph Theory Review
Compatibility relations like these must be satisfied at each and every high-symmetry point, line, and plane through-out the BZ. In particular (as discussed in the main text), in order to connect the little group representations of pairsof high-symmetry k -points, and so form global energy bands, we must ensure that the compatibility relations aresatisfied along the lines and planes joining the two points. In general, there will be many ways to form global energybands consistent with the compatibility relations, each yielding a physically distinct realizable band structure. Ourgoal is to systematically classify all these valid band structures. Since the compatibility relations are a purely group-theoretic device with meaning independent of any choice of Hamiltonian, we can accomplish this task by introducingmore refined graph-theoretic picture of band connectivity, as presented in the main text.To begin, we introduce some graph-theoretic terminology. Definition 6. A partition of a graph is a subset, V , of nodes such that no two nodes in V are connected by anedge. In our construction, each partition will correspond to a high-symmetry k -point, and irreps of the little group ofeach k -point will be represented as nodes, as shown in Fig. S1 Definition 7.
The degree of a node v in a graph is the number of edges that end on v . These definitions allow us to formalize the notion of a connectiviy graph as introduced in the main text, in particular,
Definition 8.
Given a collection of little group representations, M , (i.e. bands) forming a (physical) band represen-tation for a space group G , we construct the connectivity graph C M as follows: we associate a node, p a k i ∈ C M , inthe graph to each representation ρ a k i ∈ M of the little group G k i of every high-symmetry manifold (point, line, plane,and volume), k i . If an irrep occurs multiple times in M , there is a separate node for each occurence.The degree of each node, p a k i , is P k i · dim( ρ a k i ) , where P k i is the number of high-symmetry manifolds connected tothe point k i : dim( ρ a k i ) edges lead to each of these other k − manif olds in the graph, one for each energy band. Whenthe manifold corresponding to k i is contained within the manifold corresponding to k j , as in a high-symmetry pointthat lies on a high-symmetry line, their little groups satisfy G k j ⊂ G k i . For each node p a k i , we compute ρ a k i ↓ G k j ≈ M b ρ b k j . (S10) We then connect each node p b k j to the node p a k i with dim( ρ b k j ) edges. We give an illustration of these concepts in Fig. S1.The advantage of this graph-theoretic approach to topological phase transitions is that it is algorithmically tractable.Using the 460 tables of compatibility relations which we have generated and will publish in the accompanying Ref. 16,we have algorithmically constructed all compatiblity graphs consistent with the 5646 allowed elementary band repre-sentations, as well as for the 4757 independent physically elementary band representations. While this may naivelyseem to be a hopeless task, we have developed several algorithms, outlined in Section II C, and in more detail in theaccompanying Ref. 20, which reduce the problem to analyzing a computationally tractable ∼ . In particular, recall that Definition 9.
The adjacency matrix, A , of a graph with m nodes is an m × m matrix, where the ( ij ) ’th entry is thenumber of edges connecting node i to node j . In addition,
Definition 10.
The degree matrix, D , of a graph is a diagonal matrix whose ( ii ) ’th entry is the degree of the node i . We can then form the Laplacian matrix L ≡ D − A (S11)We make use of the following fact about the spectrum of L : Proposition 3.
For each connected component of a graph, there is a zero eigenvector of the Laplacian. Furthermore,the components of this vector are on all nodes in the connected component, and on all others. The proof of this statement follows directly from the observation that the sum of entries in any row of the Laplacianmatrix is by definition zero, coupled with the observation that if L ij = 0, then nodes i and j lie in the same connectedcomponent . We give an example of this method applied to graphene below in Section III C. ⇤ z}|{ z}|{ ¯ ¯ ¯⇤ ¯⇤ ¯⇤ ¯⇤ ¯⇤ ¯⇤ FIG. S1. Subgraph of a connectivity graph corresponding to the compatibility relations along Γ and Λ for P ¯43 m (215) asdiscussed in Sec. II A. There are two partitions in the graph labelled by Γ and Λ. In the Γ partition there are two nodesindicated by black circles, labelled ¯Γ and ¯Γ , each corresponding to a copy of the ¯Γ little group representation. Similarly,in the Λ partition, there are two nodes corresponding to copies of the ¯Λ little group representation and indicated by redcircles; two nodes corresponding to the ¯Λ representation and indicated by blue circles; and two nodes corresponding to the ¯Λ representation and indicated by green circles. The nodes are connected by edges (represented by black lines) consistent withthe compatibility relation Eq. (S9). Because there are only two partitions in this subgraph, P Γ = P Λ = 1 (c. f. Def. 8) for allnodes. The degree of each node in the Γ partition is 4 = P · dim(¯Γ ). Similarly, since dim(¯Λ )=2, the degree of the nodes ¯Λ and ¯Λ is 2. The remaining nodes in the Λ partition have degree 1, since they correspond to 1 D representations. Note, forexample, that if the Λ line was also connected to another high symmetry k -point (labelled L , for instance), then P Λ = 2, andthe degree of each node in the Λ partition would double. C. Connectivity Graphs
We apply this graph-theoretic machinery to the connectivity graphs (defined in Sec. II of the main text) associatedto elementary band representations. We start with all the little group representations at high-symmetry points andlines contained in a given EBR. Because the representation is elementary, we know that the connectivity graphs willhave either one connected component, or will decompose into a set of topological band groups, as explained in themain text. All connectivity graphs with more than one connected component, if they exist, will then correspond totopological phases.In order to construct the Laplacian matrix, we separate the task into two steps. We first construct all possibleadjacency matrices, and then we subtract the degree matrix from each of them. Since the adjacency matrices have ablock structure, with nonzero blocks determined by the compatibility relations, we first build each block submatrixseparately. We start by identifying the maximal k -vectors in the BZ. In analogy to maximal Wyckoff positions,these are the k vectors whose little co-groups are maximal subgroups of the point group of the space group. A validsubmatrix will then be created based on our derived compatibility and site-symmetry tables .The rows represent themaximal k -vectors and the columns represent the connecting (non-maximal) lines and/or planes. The entries in thesubmatrix fulfill the following rules: we can only allow one nonzero entry per column, and the sum of the entries ineach row equals the dimension of the corresponding little-group representation. Given a single valid submatrix, allothers can be obtained by permuting the columns.With these submatrices, we build up the full adjacency matrix row by row. In doing so, we must ensure thatwe account for all possible connections along non-maximal lines and planes. Additionally, we would like to avoidovercounting configurations that differ only by a relabelling of representations along non-maximal k -vectors. We havedeveloped two main tools to do this. First, although Def. 8 for the connectivity graphs makes use of all high-symmetrymanifolds in the BZ, many of them provide redundant information. We thus consider for each space group only theminimal set of paths in k -space necessary. We derived these for each space group by searching first for the paths inthe BZ connecting all maximal k -vectors along the highest symmetry surfaces possible, and then pruning connectionswhich add no additional symmetry constraints. For non-symmorphic space groups, it is also necessary to consider (a) a b b c c c (b) FIG. S2. Lattice basis vectors (a) and Wyckoff positions (b) of the hexagonal lattice. The (maximal) 1 a , 2 b and 3 c Wyckoffpositions are indicated by a black dot, blue squares, and red stars, respectively. The non-maximal 6 d and 6 e positions areindicated by purple crosses and green diamonds, respectively. The multiplicity is determined by the index of the stabilizergroup with respect to the point group C v (6 mm ). paths connecting maximal k -vectors in different unit cells of the reciprocal lattice, to account for the monodromy ofrepresentations . Second, we select from the set of valid submatrices in each block of the adjacency matrix only thosethat yield non-isomorphic connectivity graphs. A detailed discussion of the algorithm we used is given in Ref. 20.The topologically distinct connectivity graphs for each elementary band representation can be accessed through theBANDREP program on the Bilbao Crystallographic Server . III. EXAMPLE: GRAPHENE
In this Appendix, we illustrate the application of our representation- and graph-theoretic methods through theexample of graphene with spin-orbit coupling – the primordial topological insulator. We begin in Subsection III Aby reviewing the crystal structure and symmetries of the honeycomb lattice. Next, in Subsection III B we constructexplicitly the elementary band representation realized by spin-orbit coupled p z orbitals in a hexagonal lattice, andhence deduce the full symmetry content of graphene irrespective of any microscopic model . In Subsection III C weapply our connectivity graph machinery to this band representation, allowing us to catalogue the different allowedtopological phases of graphene. Finally, in Subsection III D, we show how these cases can be physically realized, andcomment on the relationship of our approach to older work. This gives a much needed practical example of how touse our formalism.. Because we are interested in spin-orbit coupled systems, for the remainder of this section we willemploy primarily double-valued point and space group representations unless otherwise specified. A. Space group symmetries
The 2 D honeycomb lattice of graphene has as its symmetry group the wallpaper group p mm (No. 17, the mostsymmetric triangular wallpaper group). This is a symmorphic group with primitive lattice basis vectors, e = √
32 ˆ x + 12 ˆ y (S12) e = √
32 ˆ x −
12 ˆ y , (S13)which are pictured in Fig S2a. Note that the Bilbao Crystallographic Server uses e = e , e = e − e (S14)as an alternative choice of primitive lattice vectors.The point group is C v , and is generated by C z :( e , e ) → ( − e , e − e ) (S15) C z :( e , e ) → ( − e , − e ) (S16) m :( e , e ) → ( e , e ) , (S17)where the subscript 1¯1 denotes that the mirror line has normal vector e − e . Although this set of generators isovercomplete (a minimal set of generators is { C z , m } ), it is convenient for our purposes. The three-dimensionalspace group with the symmetries catalogued above is space group P mm (183), which differs only in the addition ofa third translation vector; we recover the 2 D symmetry group by taking the length of this lattice vector to infinity.We note, however, that when we consider the 2 D wallpaper group as embedded in three-dimensional space, we havesome freedom when it comes to imposing extra symmetries such as inversion I . For this particular wallpaper group,we see that the combination m z = IC z fixes every point in the 2 D lattice, but acts on the spin degree of freedom asa rotation by π about the z axis. As such, imposing inversion symmetry on graphene is tantamount to imposing theconservation of S z . This will become important when we consider spin-orbit coupling. In general, however, we viewspin conservation as non-essential, and restrict ourselves in most cases to the symmetries of P mm (183).The honeycomb lattice has three maximal Wyckoff positions, as shown in Fig S2b. In graphene, the carbon atomssit at the 2 b position, with symmetry sites { q b , q b } = { (
13 13 ) , (¯ ¯ ) } . Here and throughout ¯ x = − x . The stabilizergroup G q b is isomorphic to the group C v ; it is generated by the elements { m | } and { C z | } . It is an indextwo subgroup of the point group C v , and the quotient group C v /C v is generated by the coset that contains C z (regardless of whether we are using point groups or double point groups, this quotient group is isomorphic to theabelian group with two elements, since C z = ¯ E ∈ C v ).In the BZ, we take for our primitive reciprocal-lattice basis vectors g = 2 π √
33 ˆ x + ˆ y ! (S18) g = 2 π √
33 ˆ x − ˆ y ! , (S19)which are shown in Fig S3. We will be primarily interested in the little group representations at three high symmetrypoints in the BZ. The first is the Γ point, with coordinates (00). The little co-group ¯ G Γ is, as always, the pointgroup C v . Next, there are the three time-reversal invariant M points (that is, points k such that − k ≡ k modulo areciprocal lattice vector), which we denote M , M and M . These have coordinates (
12 12 ) and (0 ) respectively.For the remainder of this appendix we need only concern ourselves with the first of these, and so we will refer to itunambiguously as “the” M point; the others are related to it by C z symmetry. It has little co-group ¯ G M , which isisomorphic to C v and generated by C z and C z m . Finally, there are the K and K points – the focus of mosttopological investigations in graphene. We will focus here primarily on the K point which has coordinates (
13 23 ); the K point can be obtained by a π/ G K is isomorphic to C v and is generated by C z and C z m . The high symmetry points are shown in Fig S3. In tables S1, S2, and S3 we give the character tablesfor the irreducible representations of the little co-groups ¯ G Γ , ¯ G K , and ¯ G M respectively. We indicate double-valued(spinor) representations with a bar over the representation label. As mentioned previously, these character tablesfully determine the representations of the corresponding little groups, since P mm (183) is symmorphic. Rep
E C z C z C z m C z m ¯ E Γ √ √ G Γ ≈ C v of the Γ point. The irreps Γ -Γ are all single valued, while¯Γ , ¯Γ , and ¯Γ are double valued. ¯Γ is the spin- representation, ¯Γ is the | S = 3 / , m z = ± / i representation, and ¯Γ is the | S = 5 / , m z = ± / i representation, all distinguishable by the action of C z . B. p z Orbitals and the Elementary band representation
In graphene, the relevant orbitals near the Fermi level are the two spin species of the p z orbitals at the 2 b Wyckoffposition. Let us focus on the orbitals {| p z ↑i , | p z ↓i } at the q b site. These transform according to an irreducible Rep
E C z C z m ¯ EK K K K K K G K ≈ C v of the K point. There are three single-valued representations K – K , and three double valued representations ¯ K – ¯ K . The one-dimensional representations ¯ K and ¯ K are complex conjugatesof each other. The two dimensional ¯ K representation is the spin- representation, while the one-dimensional ¯ K and ¯ K representations act in the space spanned by | S = 3 / , m z = 3 / i ± i | S = 3 / , m z = − / i respectively.Rep E C z m C z m ¯ EM M M M M G M ≈ C v of the M point, for both single and double-valued representations.The single-valued representations M – M are all one dimensional. The unique double-valued representation, ¯ M , is the two-dimensional spin- representation. In terms of the Pauli matrices, it is given concretely as ¯ M ( C z ) = iσ z , ¯ M ( m ) = iσ y . double-valued (spinor) representation ρ of the site symmetry group G q b = C v : in the space of these orbitals, { C z | } acts as a rotation about the z -axis in spin space, and m acts as a spin-flip. Furthermore, time-reversal symmetry T acts as a spin flip times complex conjugation. Symbolically, ρ ( { C z | } ) = e iπ/ s z , ρ ( m ) = is x , ρ ( T ) = is y K , (S20)where { s , s x , s y , s z } are Pauli matrices that act in the space of spin ↑↓ ( s is the identity matrix), and K is complexconjugation. Similarly, the p z orbitals at the q b site transform in an equivalent representation obtained by conjugationby C z .Because ρ is a physically irreducible representation of the site-symmetry group of a maximal Wyckoff position (anddoes not appear in Table S11), it induces a physically elementary band representation, ρ k G = ρ ↑ G . It has fourbands, coming from the four orbitals per unit cell. To construct the matrices ρ k G we directly examine the action ofthe point group elements on the spin and location of orbitals. Let’s focus first on { C z | } . Its action on orbitalsat the q b site can be deduced from above; for orbitals at the q b site, it acts as a rotation in spin space, and takes { C z | } q b = q b + e . Introducing a set of Pauli matrices { σ , σ x , σ y , σ z } which act in the sublattice basis ( σ is theidentity matrix), this allows us to write ρ k G ( { C z | } ) = e iπ/ s z ⊗ e i ( k · e ) σ z , (S21)where ⊗ is the usual tensor product. Similarly, m acts as a spin flip at the q b site, and also leaves this pointinvariant. Hence ρ k G ( { m | } ) = is x ⊗ σ . (S22)Next, since time-reversal acts independent of position, we know immediately that ρ k G ( T ) = is y ⊗ σ K . (S23)Lastly, we need to examine { C z | } . This interchanges the two sublattices (orbitals), and so acts as σ x in sublatticespace. In spin space, it acts as a rotation by π , and commutes with T, the time-reversal operator. Thus we deduce ρ k G ( { C z | } ) = is z ⊗ σ x . (S24)Representation matrices in hand, it is now a simple matter of comparison with Tables S1, S2, S3 to determine thelittle group representations at each high symmetry point. First, at the Γ point all point group elements are in thelittle co-group, and we see that the matrices ρ k G restrict to ρ Γ G ( { C z | } ) = e iπ/ s z ⊗ σ , ρ Γ G ( { m | } ) = is x ⊗ σ , ρ Γ G ( { C z | } ) = is z ⊗ σ x , ρ Γ G ( T ) = is y ⊗ σ K . (S25)0 FIG. S3. Reciprocal lattice basis vectors and high symmetry points of the hexagonal lattice.BR Γ
K Mρ ↑ G ¯Γ ⊕ ¯Γ ¯ K ⊕ ¯ K ⊕ ¯ K ¯ M ⊕ ¯ M TABLE S4. Little group representations for the energy bands induced from p z orbitals in graphene. Comparing the trace of each of these unitary matrices to the characters in Table S1, we see that( ρ ↑ G ) ↓ G Γ ≈ ¯Γ ⊕ ¯Γ . (S26)[Although TR is not mentioned in Table S1 or S3, the ¯Γ and ¯Γ representations of G Γ and the ¯ M representation of G M satisfy Kramers’s theorem consistent with our choice of time-reversal matrix Eq. (S23).] Next, the little co-groupat K is generated by C z and C z m which in this band representation are given by ρ KG ( { C z | } ) = e iπ/ s z ⊗ e πi/ σ z , ρ KG ( C z m ) = − is y ⊗ σ x . (S27)Upon taking traces and comparing with Table S2 we deduce( ρ ↑ G ) ↓ G K ≈ ¯ K ⊕ ¯ K ⊕ ¯ K . (S28)Finally, using the fact that there is only a unique double-valued representation ¯ M allowed at the M point, we deduceby simple dimension counting that ( ρ ↑ G ) ↓ G M ≈ ¯ M ⊕ ¯ M . (S29)Thus, we have deduced, from symmetry alone , the little group representations of the energy bands induced by the p z orbitals in graphene. For future convenience, we summarize this in Table S4. In the spirit of our programmedescribed in the main text, the next step is to analyze how these energy bands are permitted to connect throughoutthe BZ. We will accomplish this with the aid of the graph theory method outlined in Section II B. C. Graph Analysis
To construct the connectivity graphs – and hence determine the allowed topological phases of graphene, we shallfollow the procedure outlined in the main text and in Section II B. To do this, we need to first examine the compatibilityrelations along the lines joining Γ , K, and M in the BZ. Once we have determined the compatibility relations for thelittle group representations occurring in the band representation ρ ↑ G of p z orbitals, we will explicitly constructthe distinct connectivity graphs for the system. In particular, we show that there is a fully connected protected (athalf-filling) semi-metallic phase, as well as a disconnected topological insulating phase.Let us begin with the line Σ = k g , k ∈ [0 , ] which links Γ and M . The little co-group ¯ G Σ of this line is the abeliangroup C s generated by C z m . It has two double-valued representations denoted by ¯Σ and ¯Σ , distinguished bywhether the group generator is represented by ± i , respectively. Consider the little group representations appearingin the ρ ↑ G band representation in Table S4. From Table S1, we see that in both the ¯Γ and ¯Γ representations atΓ, the character χ ( C z m ) = 0 (it is in the same conjugacy class as m in the table). From this we deduce that1each of these representations restricts to the direct sum ¯Σ ⊕ ¯Σ on the line Σ. A similar analysis shows that therepresentation ¯ M at the M point subduces also to ¯Σ ⊕ ¯Σ . We summarize this in the compatibility relations (S30):¯Γ ↓ G Σ = ¯Σ ⊕ ¯Σ ¯Γ ↓ G Σ = ¯Σ ⊕ ¯Σ ¯ M ↓ G Σ = ¯Σ ⊕ ¯Σ . (S30)Next, we look at the line T = ( + k ) g + 2 k g , k ∈ [ − , K and M points. The little co-group¯ G T of this line is also isomorphic to C s , this time generated by the mirror C z m . As above, we denote its two irrepsby ¯ T and ¯ T . By looking at the characters of the little group representations ¯ K , ¯ K , and ¯ K , from Table S2, wededuce that ¯ K ↓ G T = ¯ T ¯ K ↓ G T = ¯ T ¯ K ↓ G T = ¯ T ⊕ ¯ T . (S31)The restriction of the representation ¯ M into representations of C s was computed in Eq. (S30), and so¯ M ↓ G T = ¯ T ⊕ ¯ T . (S32)Finally, we examine the line Λ = k g + 2 k g , k ∈ [0 , ], which connects the points Γ and K . Like the previous cases,the little co-group of this line is C s , this time generated by C z m , and the compatibility relations are¯ K ↓ G Λ = ¯Λ ¯ K ↓ G Λ = ¯Λ ¯ K ↓ G Λ = ¯Λ ⊕ ¯Λ ¯Γ ↓ G Λ = ¯Λ ⊕ ¯Λ ¯Γ ↓ G Λ = ¯Λ ⊕ ¯Λ . (S33)The only remaining k -surface in the BZ is the general position, denoted GP = k g + k g . However, it has atrivial little group with only one 1 D double-valued irreducible representation ¯ GP . Because of this, the compatibilityrelations are trivial – all representations restrict to copies of ¯ GP , and group theory provides no restrictions on theconnectivity. Thus this surface does not add anything new to the connectivity analysis and we omit it here.We can now construct the degree, adjacency, and Laplacian matrices, defined in Section II B, consistent with thesecompatibility relations. In each of these matrices, the rows and columns (i.e. the nodes in the connectivity graph)are labelled by the different irreps occurring in the band representation. In this example, we have the followingnodes: ¯Γ , ¯Γ , ¯Σ , ¯Σ , ¯Σ , ¯Σ , ¯Λ , ¯Λ , ¯Λ , ¯Λ , ¯ K , ¯ K , ¯ K , ¯ T , ¯ T , ¯ T , ¯ T , ¯ M , ¯ M . We reiterate here that, as per Def. 8,representations at high-symmetry points and lines correspond to nodes in our graph. Note that if a representationoccurs more than once (as is the case along the lines Σ , T , and Λ, as well as at the point M ), there is a distinct nodefor each copy that appears, which we label here with a superscript.We begin first with the degree matrix D , as defined in Def. 10. Let us denote by d ( σ ) the degree of the node labelledby representation σ in the connectivity graph. We know from Def. 8 that d ( σ ) is given by dim( σ ) times the number ofdistinct compatiblity tables in which σ appears. For example, dim(¯Γ ) = 2. Furthermore, it connects to P = 2 otherhigh-symmetry lines (in the notation of Def. 8), Σ and Λ. Following this prescription, the entry d (¯Γ ) = 2 × D which wesummarize in Table S5. ¯Γ ¯Γ ¯Σ ¯Σ ¯Σ ¯Σ ¯Λ ¯Λ ¯Λ ¯Λ ¯ K ¯ K ¯ K ¯ T ¯ T ¯ T ¯ T ¯ M ¯ M d ( ρ ) 4 4 2 2 2 2 2 2 2 2 2 2 4 2 2 2 2 4 4TABLE S5. Nonzero entries in the degree matrix for the ¯ ρ b ↑ G band representation in P mm (183) Next, we construct the allowed adjacency matrices for this band representation. The adjacency matrices all havea sparse block structure – blocks connecting different k -points are nonzero only if the points are compatible. We see2then that the only nonzero blocks are the Γ − Σ, Γ − Λ, K − Λ, K − T , M − T , and M − Σ blocks. Furthermore,up to relabellings of identical representations, i.e. ¯ M ↔ ¯ M , ¯Λ ↔ ¯Λ , etc., there are only four distinct adjacencymatrices (we elaborate on these details in Ref. 20). These fall into two groups which differ by the exchange ¯Γ ↔ ¯Γ ,since ¯Γ and ¯Γ have identical compatibility relations along both Σ and Λ. For brevity, we write here only the twoindependent matrices from which the remaining two can be obtained by the exchange ¯Γ ↔ ¯Γ . They are A = ¯Γ ¯Γ ¯Σ ¯Σ ¯Σ ¯Σ ¯Λ ¯Λ ¯Λ ¯Λ ¯ K ¯ K ¯ K ¯ T ¯ T ¯ T ¯ T ¯ M ¯ M K K K T T T T M M (S34)and A = ¯Γ ¯Γ ¯Σ ¯Σ ¯Σ ¯Σ ¯Λ ¯Λ ¯Λ ¯Λ ¯ K ¯ K ¯ K ¯ T ¯ T ¯ T ¯ T ¯ M ¯ M K K K T T T T M M (S35)These matrices differ only in their K − Λ and K − T blocks. As a consistency check, we verify that the sum of elementsin the row or column labelled by σ is equal to d ( σ ) from Table S5; thus, the degree matrix D satisfies D ij = δ ij P ‘ A i‘ .We show each of these graphs pictorially in Figure S4. Although the graph method does not impose any constraintson the energies of the irreducible representations, we are free to interpret and visualize the vertical positioning of thenodes of the graph as the energy of the respective energy bands. Doing so gives Fig. S4 the alternative interpretationas a plot of the band structure!We can now construct the Laplacian matrices L = D − A and L = D − A associated to these two graphs. Tosave space we will not write these out explicitly. We find that the null space of the matrix L is spanned by the3unique vector ψ = (cid:0) (cid:1) T (S36)indicating that the graph described by the matrix A has a single connected component consisting of all the nodes inthe graph. On the other hand, we find that the null space of L is spanned by ψ = (cid:0) (cid:1) T (S37) ψ = (cid:0) (cid:1) T (S38)indicating that the graph described by the matrix A has two connected components. Consulting our orderingof representations in Table S5, we see that the first connected component contains the little group represen-tations ¯Γ , ¯Σ , ¯Σ , ¯Λ , ¯Λ , ¯ K , ¯ K , ¯ T , ¯ T and ¯ M , while the other connected component contains the remainder¯Γ , ¯Σ , ¯Σ , ¯Λ , ¯Λ , ¯ K , ¯ T , ¯ T and ¯ M . (Interchanging ¯Γ and ¯Γ also results in a valid disconnected graph). Sinceeach of these connected components comes from splitting an elementary band representations, they each describe atopological group of bands, and hence a topological insulator.This is consistent with the results of Ref. 25, which found that the Wannier functions in the valence bands in thetopological phase of graphene were of the form | p z ↑ + ↓i localized on the A sites, and | p z ↑ − ↓i localized on the B sites (the two points in the unit cell of the 2 b Wyckoff orbit); by examining the action of C z on these orbitalswe conclude immediately that they do not, by themselves, form a representation space (carrier space) of the site-symmetry group G q b . In fact, no set of spin-1 / s or p orbitals and with one orbitalper site can respect the spatial symmetries, since C v has only two dimensional double-valued representations with m z = ± / D. Hamiltonian Analysis
We justify the preceding analysis concretely by considering a tight-binding model of p z or ( s ) orbitals centered on2 b sites with the most general Rashba and Haldane type SOC interactions. In particular, we will show how differentclasses of spin-orbit coupling terms can drive a transition between the two phases indicated in Fig S4a and S4b. Wewill use the basis of spin and sublattice (orbital) Pauli matrices (including the identity matrices) s i ⊗ σ j , i, j = 0 , , , s z , it is of “Haldane” type. All other spin-orbit coupling is of Rashba type (because any termthat breaks spin conservation in this basis is also not invariant under C z I , when inversion is taken to act in threedimensions. This is true for any two-dimensional system embedded in three dimensional space, c. f. Ref. ) Themost general Haldane-type SOC term is H HSOC ( k ) = d ( k ) s z ⊗ σ + d x ( k ) s z ⊗ σ x + d y ( k ) s z ⊗ σ y + d z ( k ) s z ⊗ σ z . (S39)Looking at the Γ point first, C z symmetry forces d y (0) = d z (0) = 0, while mirror symmetry forces d (0) = d x (0) = 0.Thus, Haldane spin orbit coupling does not affect the band structure at the Γ point. At the K point, however,Eq. (S27) shows that mirror symmetry forces d ( K ) = d x ( K ) = 0, and that C symmetry forces d y ( K ) = 0. Thus, atthe K point, Haldane spin orbit coupling takes the general form H HSOC ( K ) = λ H s z ⊗ σ z . (S40)We perform the same analysis for Rashba spin-orbit coupling. We start with the most general spin-non-conservingHamiltonian, H R ( k ) = X i = x,y X j =0 ,x,y,z d ij ( k ) s i ⊗ σ j (S41)At the Γ point, H R (0) = 0, since C z fails to commute with every term in Eq (S41) (this is perhaps well-known for thestandard Rashba term ( k × ~σ ) z , however here we have shown it is true for any spin-nonconserving term that respectsthe crystal symmetries). At the K point, however, C z symmetry allows d xy ( K ) = 0 and d yx ( K ) = 0. Furthermore,mirror symmetry forces d xy ( K ) = − d yx ( K ). Hence, H R ( K ) = λ R ( s x ⊗ σ y − s y ⊗ σ x ) (S42)4 Γ K M k E ¯Γ ¯Γ ¯Λ ¯Λ ¯Λ ¯Λ ¯ K ¯ K ¯ K ¯ T ¯ T ¯ T ¯ T ¯ M ¯ M (a) Γ K M k E ¯Γ ¯Γ ¯Λ ¯Λ ¯Λ ¯Λ ¯ K ¯ K ¯ K ¯ T ¯ T ¯ T ¯ T ¯ M ¯ M (b) FIG. S4. Band structures corresponding to the connectivity graphs for P mm (183), with little group representations alongpoints and lines labelled as shown. (a) shows the graph corresponding to the adjacency matrix A , while (b) shows the graphcorresponding to adjacency matrix A . The terms in Eqs (S40) and (S41) exhaust the space of possible SOC terms.We now analyze the effect of spin orbit coupling H H = H HSOC + H R on the band structure. We have shownalready that symmetry prohibits the spin-orbit coupling from altering the band structure at Γ. At K , the eigenvaluesof H H ( K ) are δE ± = − λ H ± λ R and δE = λ H ; the latter is two-fold degenerate, corresponding to the ¯ K littlegroup representation, while the former correspond to the ¯ K and ¯ K representations. The associated eigenvectors are ψ ± = 1 √ (cid:0) | ↑ q b i ∓ | ↓ q b i (cid:1) , ψ = | ↑ q b i , ψ = | ↓ q b i (S43)First consider the case λ R ± λ H >
0, so that δE + > δE > δE − . (We also could have chosen λ R ± λ H <
0, in whichcase δE + < δE < δE − .) The Dirac cone at K that exists without spin orbit coupling thus splits into a twofolddegenerate state sandwiched between two non-degenerate states. Since H H does not change the band structure at Γ,we will without loss of generality let the ¯Γ representation sit higher in energy than ¯Γ and the ¯ K representation sitabove ¯ K . Then the induced band representation is four-fold connected, as shown in Fig S4a. Re-ordering the bandsat Γ or swapping ¯ K and ¯ K does not change this connectivity (as noted previously in Section III C). By inspectingTable S4, we confirm that the bands transform under the four-fold connected elementary band rep ¯ ρ b ↑ G , inducedfrom G q b , as we asserted at the beginning of this section. Connecting to the graph theory analysis in Sec III C, thefour-fold band connectivity corresponds to the case of the single null vector in Eq (S36). This is an enforced semimetalphase.In the opposite regime, where sgn( λ R − λ H ) = − sgn( λ R + λ H ), the situation is more interesting. In this case, atthe K point, the twofold degenerate ψ states sit higher or lower in energy than the ψ ± states. This ordering impliesthat bands are now only twofold connected, as shown in Fig S4b. If we assume the ¯Γ representation of G Γ sits higherin energy than ¯Γ and λ R − λ H < λ R + λ H > irrep of G Γ (since H HSOC (0) = 0), while conduction bands at the K point transform under the ¯ K representationof G K . Additionally, the valence bands transform under the ¯Γ irrep at the Γ point and the ¯ K ⊕ ¯ K irrep at the K point. Thus, while it is true that all four bands together still transform under the ¯Γ ↑ G band representation inducedfrom G q b , for these parameters, the valence bands and the conduction bands alone do not form a band representation.This possibility was introduced in the graph theory analysis in Sec III C and the two disconnected pieces correspondto the two null vectors in Eqs (S37) and (S38). IV. HYBRIDIZATION AND TOPOLOGY IN THE 1D CHAIN
To elucidate the connection between hybridization, bonding, and topological phases, we will here examine a simple,well known model in a new light: a 1D inversion symmetric chain with one (spinless) s orbital and one (spinless) p x orbital per site. While formally equivalent to the Su-Schrieffer-Heeger and Rice-Mele models, in this formulationthe connection of topology and chemistry is manifested.We begin by defining our lattice, a schematic diagram of which is shown in Fig. S5a. Lattice sites in the figure arerepresented by black circles. We take as our origin the lattice site within the green dashed rectangle, which denotes5 b a c c ~t (a) ± = { (b) FIG. S5. The 1D inversion-symmetric chain. (a) shows a schematic diagram of the 1D inversion symmetric lattice, spacegroup p ¯1. The lattice sites are shown in black circles, and the Bravais lattice translation vector is labelled by ~t . The greendashed square outlines a single unit cell of the lattice. The lattice site itself serves as our choice of inversion center, and is the 1 a Wyckoff position. The blue square at the edge of the unit cell is the 1 b Wyckoff position. The two red stars indicate the pointsin orbit of the non-maximal 2 c Wyckoff position. (b) is a schematic representation of sp -hybridized orbitals relevant to thetransition between the trivial and topological phases. They are obtained as symmetric and antisymmetric linear combinationof s and p orbitals. one unit cell. With this choice of origin, The space group G is generated by G = h{ E | ~t } , { I | } , T i , (S44)where T is spinless time reversal (although time-reversal is not necessary for the following discussion, we include ithere to limit the number of allowed terms in the Hamiltonian). There are three distinct Wyckoff positions in theunit cell of this crystal. The first, labelled 1 a , has coordinates q a = 0, located at the inversion center. The sitesymmetry group G q a is generated by { I | } and T . This is a maximal Wyckoff position, since the site symmetry groupis isomorphic to the point group of p ¯1. Similarly, the second maximal position is labelled 1 b , and has coordinates q b = in units of the lattice constant. The site-symmetry group G q b is also isomorphic to the point group ¯1, butnow generated by { I | } and T .Finally, there is also the non-maximal Wyckoff position 2 c , with coordinates { q c , q c } = { x, − x } . The stabilizergroup of either of these sites contains only time-reversal and the identity element.In reciprocal space, the BZ of the crystal is an interval generated by the reciprocal lattice translation ~g satisfying ~g · ~t = 2 π . In units of ~g , there are two inversion symmetric points in the BZ: Γ = 0 and X = ( ≡ π ).Now, we enumerate the elementary band representations for this space group. First, we note that the full pointgroup – and hence the site-symmetry groups G q a and G q b – have two one-dimensional irreducible representations ρ ± distinguished by whether or not the inversion element is represented by ±
1; in both cases time-reversal is representedby complex conjugation. We can carry out the induction procedure for each maximal Wyckoff position. Becausethe generating element of G q a contains no translation, the inversion matrix is momentum-independent in bandrepresentations induced from this site, and so the inversion eigenvalues at Γ and X are identical. Conversely, becausethe generating element of G q b contains a lattice translation, the inversion eigenvalues at Γ and X differ in parity inband representations induced from this site. We summarize the results in Table S6.From the table, we note that the composite band representations ( ρ a + ⊕ ρ a − ) ↑ G and ( ρ b + ⊕ ρ b − ) ↑ G have thesame representation content in momentum space. In fact, these composite band representations are equivalent: Theintersection G q a ∩ G q b = G q c contains only identity and time-reversal. The unique irrep of this group induces therepresentations ρ a + ⊕ ρ a − at the 1 a position, and ρ b + ⊕ ρ b − at the 1 b position (this follows from the fact that theseare the unique two-dimensional site-symmetry representations with zero character of inversion, the so-called regularrepresentations ) Hence this is an equivalence of band representations, which we write as( ρ a + ⊕ ρ a − ) ↑ G ≈ ( ρ b + ⊕ ρ b − ) ↑ G. (S45)Now let us consider the band structure induced by a single s and single p x orbital at the 1 a site of the lattice.Because s orbitals transform in the ρ a + representation, while p orbitals transform in the ρ a − representation, the full6 Position Rep Γ X a ρ a + ↑ G + + ρ a − ↑ G − − b ρ b + ↑ G + − ρ b − ↑ G − +TABLE S6. Elementary band representations for the 1D space group p ¯1. The first column indicates the Wyckoff position, andthe second column the band representation induced from this site. The third column gives the eigenvalue of inversion at the Γpoint in the BZ. The last column gives the eigenvalue of inversion at the X point. two-band band structure transforms in the ( ρ a + ⊕ ρ a − ) ↑ G band representation. Keeping in mind the equivalenceEq. (S45), this means that Wannier functions for the two bands taken together can lie anywhere along the linebetween the 1 a and 1 b position. If the energetics are such that the system is gapped (which is generically the case in1D), then the exponentially localized valence band (which is fully occupied at a filling of one electron per unit cell)Wannier functions will lie either on the 1 a or 1 b site.To see this concretely, let us construct an explicit tight binding model consistent with these symmetries. The mostgeneral nearest-neighbor Bloch Hamiltonian induced from an s and a p orbital on the 1 a site takes the form H ( k ) = − [ (cid:15) + ( t ss + t pp ) cos( ka )] σ z − t sp sin( ka ) σ y , (S46)where a is the lattice constant, (cid:15) is the onsite-energy difference between s and p orbitals, t ss is s − s hopping, t pp is p − p hopping, and t sp is the interorbital hopping (had we chosen to break time-reversal symmetry, there would bean additional allowed s − p hopping term, which does not affect the fundamental physics). This Hamiltonian mapsto the Su-Schrieffer-Heeger model under a rotation σ z → σ x . In this basis, the inversion matrix is represented as ρ k ( { I | } ) = σ z . (S47)There are two simple limits of this model which we will analyze. First, consider the case where t ss = t sp = t pp = 0.In this limit, the spectrum is gapped and given by E ± ( k ) = ± (cid:15). (S48)This limit corresponds to decoupled atoms, and so we expect the valence band Wannier functions to be localized onthe 1 a sites. We can verify this by computing the Wannier center polarization from the Zak phase . Indeed, theoccupied band eigenfunction ψ ( k ) = (cid:18) (cid:19) (S49)is k -independent and periodic, so the Zak phase is 0. From this we deduce that the Wannier functions are localizedat the origin of the unit cell, which here is the 1 a site. We find the same result for the conduction band. In this phasethen the Wannier functions are nearly identical to the original atomic orbitals. To see this another way, comparingwith Eq. (S47) shows that the valence band transforms in the ρ a + ↑ G band representation, and so the occupiedWannier functions are s -like orbitals localized at the 1 a site.Now let us consider the opposite limit (cid:15) = 0 , t ss = t pp = t sp = t . We find that the spectrum is also flat, with E ± = ± t, (S50)However now the valence band eigenfunctions are nontrivial. Since the Hamiltonian is a sum of Pauli matrices, wecan write immediately that ψ − ( k ) = e ika/ (cid:18) cos ka i sin ka (cid:19) , (S51)where we have included a prefactor to make the wavefunction periodic. We find for the Zak phase φ = i Z π − π dkψ †− ∇ k ψ − = π, (S52)7 (a) (b) X Γ X k E (c) FIG. S6. Spectra for the 1 D inversion symmetric chain in finite and infinite size. Because we included only nearest-neighborhopping and neglected a constant on-site energy in the Hamiltonian (S46), there is an additional inessential particle-holesymmetry in the spectrum. (a) shows the spectrum for a chain of 100 sites in the trivial phase with (cid:15) = 0 . t ss = t sp = t pp = 0 .
2; note that the spectrum is fully gapped. (b) shows the spectrum for a chain of 100 sites in the topological phase with t ss = t sp = t pp = 0 .
45 and (cid:15) = 0 .
4. There are a pair of topological edge states, one localized on either edge of the chain. (c)shows the bulk spectrum, which is identical in the two cases. from which we deduce that the valence band Wannier functions are localized a half-translation from the origin ofthe unit cell, which here is at the 1 b site. We draw the same conclusion by examining the conduction band Wannierfunctions. Additionally, comparing with Eq. (S47), we see that the occupied band Wannier functions transform inthe ρ b + ↑ G band representation, and so are s -like orbitals centered on the 1 b site. This is what is routinely called the“topological phase” of the 1 D chain. However, based on our Definition 1 in the main text, it represents just anotheratomic limit; it can be described by symmetric, localized Wannier functions. Between the two atomic limits, there isa phase transition, and hence an “edge mode,” as was first pointed out by Shockley . The nonzero polarization wehave computed is the bulk signature of this edge mode. We show this for generic parameter values in Figs. S6a-S6c.This proves that edge modes can occur at the interface between distinct atomic limits, however they can be pushedinto the bulk spectrum in these cases via an appropriate edge potential.As they are localized at the center of the unit cell, the Wannier functions in the topological phase clearly do notderive from the original atomic orbitals. How then, are we to understand them? The answer comes from chemistry.Given an s and p atomic orbital with nearly the same energy (i.e. (cid:15) ≈ sp hybrid orbitals ,as shown in Fig. S5b. When the hopping amplitudes t are also large compared to (cid:15) , chemical theory tells us we shouldwork in the basis of molecular bonding and antibonding orbitals formed by taking linear combinations of sp orbitalson adjacent atoms. These molecular orbitals are also inversion symmetric, and therefore lie exactly halfway betweenthe atoms, at the 1 b site ! We thus see that the topological phase transition between trivial and nontrivial phasesof the chain is a chemical transition between weak and strong covalent bonding, where the formation of molecularbonding orbitals leads to a quantized charge polarization and edge states. V. SURVEY OF MATERIAL PREDICTIONS
With our theory now developed, we move on to apply our method to find new topological materials. The strategyfor this search has already been presented in the main text. Here we summarize our early findings. For topological in-sulators, we have identified new, broad classes of materials. The first, Cu ABX , with A=Ge,Sn,Sb, B=Zn,Cd,Hg,Cu,and X=S,Se,Te, was introduced in the main text. There are a total of 36 materials in this structure type. Thesecompounds are analyzed in detail in Subsection V A.In Subsection V B, we examine a large class of layered materials with square nets of As, Bi, and Sb. These fallinto two space groups. First, in P /nmm (129) there is the class WHM with W=Ti,Zr,Hf, or a rare earth metal,H=Si,Ge,Sn,Pb, and M=O,S,Se,Te as well as the class ACuX with A a rare earth metal and X=P,As,Sb,Bi. A smallnumber of materials in these families have been shown to be topological before , however here we will presenta general group-theoretic argument for why they all must be topological generically. These arguments additionally8 Γ M K Γ A L H A|L M|K H -6-4-2024 E n e r gy ( e V ) IrTe2soc E F (a) Γ M K Γ A L H A|L M|K H -4-202 E n e r gy ( e V ) soc E F (b) Γ X M Γ R X|M R -4-3-2-1012 E n e r gy ( e V ) soc E F (c) Γ X M Γ R X|M R -4-3-2-1012 E n e r gy ( e V ) soc_pz E F (d) Γ H N Γ P H|P N -0.4-0.200.20.4 E n e r gy ( e V ) soc E F (e) FIG. S7. Band structures for new topological insulators and semimetals. (a) shows the band structure for IrTe in P ¯3 m in the same space group, with the topologically nontrivial valence bands shown in red. (c) gives theband structure for unstrained Pb O in
P n ¯3 m (224). The isolated group of bands near − . O under uniaxial strain, which opens a topological gap nearthe Fermi level. Finally, (e) gives the band structure for Cu TeO in Ia ¯3 (206). The twenty-four bands at the Fermi level inthis material are half filled, and form the highest-dimensional PEBR allowed for any of the 230 space groups. allow us to identify 58 new topological insulator candidates in the distorted P nma (62): LaSbTe , SrZnSb , andAAgX with A a rare earth metal and X=P,As,Sb,Bi.In Subsection V C we present realizations of sixteen-fold connected metals, where crystal symmetries force sixteenbands to be connected throughout the BZ. These metals can realize exotic filling fractions (7 / , NiTe , and HfTe in P ¯3 m , here we have used our powerful connectivity theoryto find candidate materials with Dirac points at or very near the Fermi level, as shown in Fig. S7a. Also in this spacegroup, we identify CNb as a promising topological insulator candidate. We show its band structure in Fig. S7b.Additionally, we have identified topological bands below the Fermi level in Pb O in P n ¯3 m (224), shown in Fig. S7c.Furthermore, we predict that under uniaxial strain in the z -direction, the strucure distorts to P /nnm (134), anda topological gap opens near the Fermi level. This is shown in Fig. S7d. Lastly, we find a candidate for a , Cu TeO , in Ia ¯3 (206). In this material, a twenty-four band EBR is half-filled at the Fermi level, realizing the most interconnected EBR allowed by symmetry. We show the band structurein Fig. S7e. Additional candidates for exotic metals can be found in Table S15. A. Cu ABX The Cu ABX materials all belong to the symmorphic tetragonal space group I¯42m (121). This group is body-centered, and so we take for a basis of lattice vectors e = 12 ( − a ˆ x + a ˆ y + c ˆ z ) , e = 12 ( a ˆ x − a ˆ y + c ˆ z ) , e = 12 ( a ˆ x + a ˆ y − c ˆ z ) . (S53)9In addition to these translations, the space group is generated by a fourfold roto-inversion IC z ≡ S − about the z -axis, and the rotation C x about the x -axis. There are four maximal Wyckoff positions, labelled 2 a, b , 4 c, and 4 d (divided by 2 for the primitive cell description given in Eq. (S53)). The coordinate triplets of the symmetry equivalentpoints in the unit cell, with respect to Eq. (S53), are given by: q a = (0 , , , (S54) q b = ( 12 , , , (S55) { q c , q c } = { ( 12 , ,
12 ) , (0 , ,
12 ) } , (S56) { q d , q d } = { ( 34 , ,
12 ) , ( 14 , ,
12 ) } . (S57)with stabilizer groups G q a ≈ G q b ≈ D d (S58) G q c ≈ D (S59) G q d ≈ S . (S60)We note that the 4 c position with site-symmetry group D is exceptional as per Table S11, althoug this will not playa role here. We also will need to consider the non-maximal 8 i Wyckoff position, with coordinates { q ij } = { ( x + z, x + z, x ) , ( z − x, z − x, − x ) , ( − x − z, x − z, , ( x − z, − x − z, } (S61)and stabilizer group C s , generated by a single mirror. As this group is a proper subgroup of the stabilizers G q a and G q b , composite band representations induced from the 8 i can be labelled by sums of elementary band representationsfrom either the 2 a or 2 b positions. We show the crystal structure for these compounds in Fig. S8. Furthermore, thecharacter table for the group D d is shown in Table S7. We will also need the repsresntations of G q d ≈ S . Since thisis an abelian group generated by the single element IC z , all of its representations are one dimensional, and specifiedby the character χ ( IC z ). We list these in Table S8 below. Rep
E C z IC z C x m ¯ Eρ a ρ a − − ρ a − − ρ a − − ρ a − ρ a −√ − ρ a √ − D d , which is the stabilizer group of both the 2 a and 2 b positions in I ¯42 m (121) Rep IC z ρ d ρ d − ρ d iρ d − i ¯ ρ d e πi/ ¯ ρ d e πi/ ¯ ρ d e πi/ ¯ ρ d e iπ/ TABLE S8. Character table for the point group S , which is the stabilizer group of the 4 d position in I ¯42 m (121) In the particular compounds of interest, the Cu atoms sit at the 4 d position, the A atoms sit at 2 b , the B atomsat 2 a , and the X atoms at 8 i . By consulting Section VI, we see that the elementary band representations induced0 CuABX
FIG. S8. Crystal structure of the Cu ABX class of compounds, with A=Ge,Sn,Sb, B=Zn,Cd,Hg,Cu, and X=S,Se,Te. Theblue circles represent the Cu atoms at the 4 d position, which contribute at the Fermi level.FIG. S9. Band structure for the topologically trivial insulator Cu GeZnS with spin orbit coupling included. The valence andconduction bands each form separate physical band representations, and so the 0 . eV band gap is topologically trivial. from the one-dimensional representations of the stabilizer group of the 4 d position respect time-reversal symmetryin momentum space. Because of this, we know from Section I that the physically elementary band representationsinduced from this site can be disconnected, and hence topological. Furthermore, ab-initio calculations reveal that inthis material class, the relevant states near the Fermi level come from d orbitals at the 4 d position, and p orbitals atthe 8 i position. Our discussion in the main text thus flags this group of materials as prime candidates for topologicalinsulators.We focus below on three cases out of this large class of 36 materials. First, there is Cu GeZnS . Ab initiocalculations reveal this to be a large gap (trivial) insulator without spin-orbit coupling, and hence it will remainso for weak SOC. Indeed, we find that the eighty-four valence bands nearest the Fermi level transform accordingto the physical composite band representation (6¯ ρ b ⊕ ρ b ⊕ ρ d ⊕ ρ d ⊕ ρ d ⊕ ρ d ) ↑ G , while the lowest lyingconduction band transforms according to the physically elementary ¯ ρ b ↑ G band representation. In this case, all bandrepresentations induced from the 1 D representations of G q d are ”occupied”, and hence the material is topologicallytrivial. We show the band structure for this material in Fig. S9Instead, let us consider Cu SbCuS . Without SOC, this material is a zero-gap semimetal. Furthermore, whenSOC is included, the ¯ ρ d ↑ G band representation and the ¯ ρ b ↑ G representation are exchanged between the valenceand conduction band as compared with Cu GeZnS . As such, the conduction band of Cu SbCuS consists of the¯ ρ d ↑ G band representation, induced from the one-dimensional ¯ ρ d site symmetry representation. Thus, this bandrepresentation is elementary, but not physically elementary. We conclude that this material is a topological insulator.1 Γ X M Γ Z A R Z -2-1.5-1-0.500.51 E n e r gy ( e V ) Cu3SbS4 E F Γ -0.2-0.100.10.2 E n e r gy ( e V ) zoom E F π -0.500.5 θ ( π ) FIG. S10. Band structure and Wilson loop for the topologically nontrivial compound Cu SbCuS . The conduction band heredoes not form a physically elementary BR induced from a one-dimensional site-symmetry representation. The left panel showsthe band structure, with inset showing a zoomed in view of the gap at Γ. The right panel shows the calculated Wilson loopspectrum. The winding of the Wilson loop shows that this material is a strong topological insulator. k z = 0 k z = ⇡ FIG. S11. Band structure and Wilson loop for the topologically nontrivial compound Cu SnHgSe . The left panel shows theband structure, with the inset showing a zoomed in view of the (rather small) gap at the Γ point. The right panel showsthe Wilson loop calculated and k z = 0 and k z = π ; the winding of the Wilson loops reveals that this compound is a strongtopological insulator. We show the band structure for this material in the left panel of Fig. S10. To confirm our group-theoretic result,we have computed the Wilson loop spectrum, shown in the right panel of Fig. S10. The spectrum clearly windsnontrivially throughout the BZ, indicating that Cu SbCuS is a strong topological insulator. Note also that thereal-space time-reversal partner ¯ ρ d ↑ G band representation is 0 . eV below the Fermi level, although the gap in thematerial is only 0 . eV . The two time-reversed partner EBRs are shown in red in the inset of Fig. S10. Hence a novelfeature of this material is that the “topological gap” is much larger than the transport gap.Finally, we find Cu SnHgSe . We show its band structure and Wilson loop in Fig. S11. It also is a zero-gapsemiconductor without SOC which becomes a strong-topological insulator when SOC is turned on. This particularstrong TI, however, is not distinguishable from its purely group-theoretic properties: its valence and conductionbands have the same little group representations at every k point as true physical band representations; however theyare topologically nontrivial. To see this, we can calculate their Wilson loop (Berry phase, holonomy), and from itdetermine any nontrivial topological indices . Our discovery of this new TI further highlights the power of ourmaterials search.2 e e FIG. S12. Maximal Wyckoff positions in the square net. The blue star indicates the a position at the 2D lattice sites, the reddiamond indicates the b position at the center of the square cell, and the black circles denote the c Wyckoff position at themiddle of the edges.
B. Square net topological insulators
Next, we look at topological insulators of the type (1 ,
2) as defined in the main text. These materials are enforcedsemimetals with a single partially filled elementary band representation without SOC, which then splits into a topo-logically disconnected composite band representation when spin-orbit coupling is included. We consider square netsof As, Sb, Sn, and Bi which form layered compounds in P /nmm (129) and P nma (62) (upon small distortion of thesquares). We find approximately 400 candidate materials of these types, discovered by targeting our method towardsthe specific cases of orbitals which can create topological bands. In each of these classes, the relevant states near theFermi level come from the p -orbitals of the square-net atoms. The maximal positions within the square net layer arestill those shown in Fig. S12. Representative crystal structures for these compounds are shown in Figure S13. a = b → I mmm a = b → P nmma ≠ b → Pmmn a ≠ b → Pnma
CaMnBi2SrMnBi2 ZrSnTe SrZnSb2Bi2SrBi1Mn Bi2CaMnBi1 SnTeZr SrZnSb1Sb2
FIG. S13. Crystal structures for the Bi-square net class of topological insulators. The first and second structures show CaMnBi and ZrSnTe in space group P /nmm (129). In CaMnBi the Bi2 atoms form the square net, while in ZrSnTe it is the Sn atoms.The third structure shows SrZnSb in P nma (62). Here it is the atoms labelled Sb2 which make up the slightly distorted squarenet.
To analyze these materials, we first begin without SOC. Viewing the square net in isolation, we find that the Fermilevel sits between the p z orbital bonding and anti-bonding states, as shown for Bi in Figure S14. However, chargetransfer of two electrons per unit cell from the adjacent non-square net layers shown in Fig. S13 for each of thesematerials fill the p z antibonding states, putting them below the Fermi level; at the Fermi level, the { p x , p y } bondingstates are filled, while the antibonding states are empty. However, in these materials, the { p x , p y } bonding andantibonding states form a single, connected four (per-spin) band PEBR. Thus, the band structure of each quasi-2D3layer has at the Fermi level a single half-filled elementary band representation without SOC, coming from the four { p x , p y } orbitals per unit cell. This band representation is induced from the two-dimensional representation of the site-symmetry group D d , as indicated in Table S13; recall that the character table for this group was given in Table S7.This site-symmetry representation is spanned by p x ± ip y orbitals. The band structure for this band representationin a square net of Bi − ions is shown in Figure S15a. Note that at half-filling, there is a linear band crossing alongthe Γ − M line, which is the cross-section of a line-node (line-degeneracy) protected by mirror symmetry. p x , p y p x , p y p z p z Bi Bi
FIG. S14. Crystal field splitting of levels in the Bi square net. For undoped bismuth with three electrons per atom, the Fermilevel sits at the blue dotted-dashed line. Note that the four { p x , p y } states transform in a single elementary band representation. This line node is key to the topological nontriviality of these materials when SOC is included. From Table S14, wesee that with SOC the { p x , p y } orbitals decompose into the reducible ¯ ρ ⊕ ¯ ρ representation of D d , and hence inducea physically composite band representation. Note that the ¯ ρ representation is spanned by {| p x + ip y , ↑i , | p x − ip y , ↓i} states, while ¯ ρ is spanned by the {| p x + ip y , ↓i , | p x − ip y , ↑i} . Thus, for these two physically elementary bandrepresentations (¯ ρ ↑ G and ¯ ρ ↑ G ) to separate in energy and give a trivial insulator, SOC must be large enough tocompletely separate the initially degenerate spin-up and spin-down states in Fig. S15b. D d Γ M X
Σ (Γ- M ) ∆ (Γ- X ) d ρ ↑ G Γ +5 ⊕ Γ − M ⊕ M X ⊕ X Σ ⊕ Σ ⊕ Σ ⊕ Σ ∆ ⊕ ∆ ⊕ ∆ ⊕ ∆ ρ ↑ G ¯Γ ⊕ ¯Γ ¯ M ¯ X ⊕ ¯ X ρ ↑ G ¯Γ ⊕ ¯Γ ¯ M ¯ X ⊕ ¯ X p -orbitals in a square net, both with and without spin-orbit coupling. Note thatthe double-valued band representations are distinguished by the little-group representations they subduce at Γ. The dimensionsof the representations d are shown in the last column of the table However, even arbitrarily small spin orbit coupling will gap the aforementioned line node seen along Γ − M .In contrast to the trivial gap, this gap is topological – neither the valence nor the conduction band transform aselementary band representations. To see this concretely, let us examine the case of P /nmm (129). In Table S9we give the little group representations at each high-symmetry point arising from the band representations inducedby { p x , p y } orbitals; ρ is the two-dimensional SOC-free representation of D d , while ¯ ρ and ¯ ρ are the two relevanttwo-dimensional double-valued representations. The key is that without spin orbit coupling, the Γ +5 and M littlegroup representations at Γ and M respectively ”lie” in the valence band, while the Γ − and M representations ”lie” inthe conduction band. Next, we note that for arbitrarily small spin-orbit coupling, the spin representations decomposeas Γ +5 → ¯Γ ⊕ ¯Γ (S62)Γ − → ¯Γ ⊕ ¯Γ (S63) M → ¯ M (S64) M → ¯ M . (S65)We thus see that with weak spin-orbit coupling, the valence band contains the ¯Γ and ¯Γ little group representationsat Γ. However, comparing with Table S9, we see that this is not possible if the valence band is a physically elementaryband representation. While this particular energy ordering was determined from ab-initio calculations, we see that4 (a) (b) FIG. S15. Representative band structure for the bands in the Bi square net induced from { p x , p y } orbitals. (a) shows theband structure without SOC, showing band crossings at the Fermi level. These gap with infinitesimal SOC into a topologicallynontrivial insulator, as shown in (b) Γ X S Y Γ Z -3-2-101 E n e r gy ( e V ) ZrSnSb2 E F (a) Γ X S Y Γ Z U R T Z -5-4-3-2-1012 E n e r gy ( e V ) LaSbTe E F (b) FIG. S16. Representative band structures for new topologically nontrivial insulators in the distorted Bi- square net structuregroup. (a) shows the band structure of the 3 D weak topological insulator SrZnSb , while (b) shows the band structure of the3D weak topological insulator LaSbTe. the same analysis holds whenever there is one occupied and one unoccupied little group representation at Γ withoutSOC; this is generically true at half-filling. We thus deduce that for small spin-orbit coupling, these materials aretopological insulators.The ubiquity of the square net structure in nature allows us to identify hundreds of topological insulators inthis class. In space group P /nmm (129) we find materials in the class of ABX , with A a rare earth metal,B=Cu,Ag and X=Bi,As,Sb,P, for a total of 48 candidate materials. Furthermore, the recently discovered topologicalphase in tetragonal bismuth falls into this class of square-net topological insulators [albeit in I /mmm (139)].Additionally, in P /nmm (129) we find square-net compounds of the type ABX with A=Ti,Zr,Hf, or another rareearth, B=Si,Ge,Sn,Pb, and X=Os,S,Se,Te. In total, this yields 328 candidate materials in this space group.
1. Distored Square Nets
Although our analysis has focused primarily on the idealized square net, we can show that topological behavior isinsensitive to lattice distortions. We can see this most clearly by examining crystal structures with distorted squarenets. In particular, we focus on
P nma (62), which is obtained from the idealized square net in P /nmm (129)after an in-plane C symmetry-breaking distortion, shown schematically in Fig. S13. We find the 58 new candidatetopological insulators LaSbTe, SrZnSb , and AAgX , for A a rare-earth metal and X=P,As,Sb,Bi. Representative bandstructures are shown in Fig S16, where the topological gap can be clearly seen. We expect all these materials to sharea qualitatively similar topological band structure. We note empirically that the magnitude of this distortion appearsto be inversely correlated with the strength of spin-orbit-coupling of the atoms in the square net. We conjecture thatthis is due to the fact that SOC alone lifts the electronic degeneracy that causes the distortion through the Jahn-Teller5 FIG. S17. Crystal structure of the A B class of materials in I ¯43 d (220). One conventional unit cell is shown. The smallgreen circles indicate the location of the A atoms at the 12 a and 48 e Wyckoff position. The larger purple circles indicate theB atoms at the 16 c Wyckoff positon. effect.
C. Sixteen-fold connected metals
Space group I¯43d (220) supports a sixteen-band physically elementary band representation. In any topologicallytrivial phase, all sixteen of these bands need to be connected. We believe this set of high-connectivity bands, farexceeding the minimum connectivity of Refs. 46 and 47, can lead to robust protected semimetals with large conduc-tivities, strong correlations, Mott physics, and other exotic properties. We find examples of this band representation,partially filled at the Fermi-level, in the series of compounds A B , with A=Cu,Li,Na and B=Si,Ge,Sn,Pb. It isamusing to note that these materials which we identified with group theory, are also known as promising candidatesfor the next generation of batteries . Thus, batteries seem to be symmetry-protected (semi-)metals. The crystalstructure for these compounds in a conventional unit cell is shown in Figure S17. Note that there are two formulaunits per primitive unit cell.To analyze these materials, we first review the basic facts about space group I ¯43 d (220). This is a non-symmorphic,body centered cubic space group. We take as primitive basis vectors for the BCC lattice e = a − ˆ x + ˆ y + ˆ z ) , e = a x − ˆ y + ˆ z ) , e = a x + ˆ y − ˆ z ) , (S66)which we recognize as Eq. (S53) with the lattice constants a = c . In addition to the translations, space group I ¯43 d (220) is generated by the cubic threefold rotation { C , | } about the [111] axis, the four-fold roto-inversion { IC , | } about the ˆ x = e + e axis, and the mirror { m |
12 12 12 } that sends ˆx ↔ ˆy .There are three maximal Wyckoff positions in this space group, denoted 12 a, b and 16 c , with multiplicity 6 , e Wyckoff position,with multiplicity 24. The A atoms sit at the 12 a and 48 e positions, while the B atoms sit at the 16 c position. Sincethe electrons near the Fermi energy come from the B atoms, we will here be interested only in the 16 c position. It hasrepresentative coordinate q c = ( x, x, x ) in terms of the lattice vectors Eq. (S66); it is clear that the stabilizer groupis G q c ≈ C , generated by the threefold rotation { C , | } . The coordinate triplets of its symmetry equivalentpoints in the primitive unit cell are obtained by the repeated action of { IC , | } and { m |
12 12 12 } . Because thestabilizer group C is abelian, its double-valued representations are all one-dimensional, and specified by the character χ ( { C , | } ). The three possible double-valued representations are¯ ρ c ( { C , | } ) = − , (S67)¯ ρ c ( { C , | } ) = e − iπ/ , (S68)¯ ρ c ( { C , | } ) = e iπ/ . (S69)Consulting Section VI, we see that in the physically elementary band representation (¯ ρ c ⊕ ¯ ρ c ) ↑ G , Kramers’stheorem forces connection between bands coming from the ¯ ρ c ↑ G and ¯ ρ c ↑ G (non-physically) elementary bandrepresentations. As such, in any trivial phase, this band representation is sixteen-fold connected.In the A B class of materials, the B atoms sit at the 16 c Wyckoff position. In the particular examples of Cu Si ,Li Ge , Li Si , Na Sn , and Na Pb ,the relevant states at the Fermi level are precisely the B atom p -states, ofwhich there are 48 per unit cell. Due to charge transfer with the A atoms, there are 46 electrons filling these states.32 out of those 46 electrons go into filled valence bands, leaving 14 electrons to fill a band of connectivity 16. FromTable S14, we see that these yield bands transforming in the (2¯ ρ c ⊕ ρ c ⊕ ρ c ) ↑ G composite band representation.6 Γ H N Γ P H -2-1012 E n e r gy ( e V ) Cu15Si4 E F (a) Γ H N Γ P H -3-2-1012 E n e r gy ( e V ) soc E F (b) Γ H N Γ P H-3-2-1012 E n e r gy ( e V ) soc E F (c) FIG. S18. Band structures with spin-orbit coupling for the sixteen-fold connected metals in the A B structure group. Spin-orbit coupling has been included in all calculations. (a) shows the band structure for Cu Si ; the Cu d-orbitals can be seenfar below the Fermi level. (b) shows the band structure for Li Ge . Finally, (c) shows the band structure for Na Pb . In the materials listed above, ab-initio calculations reveal that the band representation closest to the Fermi-level isprecisely the sixteen-branched (¯ ρ c ⊕ ¯ ρ c ) ↑ G band representation, which by electron counting is 14 /
16 = 7 / P point these materials host a threefold degenerate fermion, while at the H point they have eightfold degenerateexcitations. In fact, our site-symmetry tables of Ref. 16 reveal that the Kramers-enforced connection between the¯ ρ c ↑ G and ¯ ρ c ↑ G band representations occurs precisely at an eightfold degeneracy point at H . D. Twenty-fourfold connected metals
An exhaustive search of the dimensions of all 10,403 EBRs and PEBRs shows that the greatest number of bands thatare forced to be connected by symmetry in a topologically trivial phase is 24. An example of this occurs in Ia ¯3 (206).In this group the 24 d maximal Wyckoff position has multiplicity two, and site-symmetry group isomorphic to C . Inspin-orbit coupled systems with TR symmetry, the two-dimensional physically irreducible ¯Γ ⊕ ¯Γ representation ofthis site-symmetry group thus induces a twenty-four band PEBR. In any topologically trivial phase, all twenty-fourof these bands must be interconnected. In Fig. S7e we show the band structure of Cu TeO , which we calculate, hasthis PEBR half-filled at the Fermi level. Although interaction effects cause this material to be a Mott insulator ,expect other materials with this PEBR near the Fermi level may exhibit exotic fillings due to charge-transfer effects.7 VI. SUPPLEMENTARY DATA
Site symmetry group Reducing group intersection group Rep dimension Space Group Number( G q ) ( G q ) ( G ) D C i C , , , , T h C O C T C , , C h C , , , D C h C D O C , , C h C , D d D h C v T h C v T d C v , , D h C v , , , , composite band representations for the single groups and thus do not needto be considered in a search for elementary band reps; computed by Bacry, Michel, and Zak . Point group symbols are given inSchoenflies notation. The first column gives the maximal site-symmetry group, G q , which induces the composite representation.The second column gives the site-symmetry group, G q , into whose band representations this composite representation can bereduced. The third column gives the intersection group, G = G q ∩ G q . The fourth column gives the dimension of the irrepwhich induces the composite band rep. The fifth column indicates the space groups for which this occurs. With (spinless)time-reversal, only the groups below the double line yield composite physical band representations (and do not need to beconsidered in a search for physically elementary band reps). Site symmetry group Reducing group Intersection group Rep dimension Space Group Number( G q ) ( G q ) ( G ) T d D d C v , ∗ O h C v D T h C O C T C , , C h C ∗ , ∗ , ∗ , ∗ C i C ∗ , ∗ , ∗ , ∗ , ∗ D h D d C v ∗ , ∗ D C h C ∗ D O C , , C h C ∗ , ∗ C v C v C s C v C s C h C s ∗ , ∗ , ∗ , ∗ , ∗ C v C s , D d C s , D T C , , , , , D C , D C , , , , , S C ∗ , ∗ , ∗ , , ∗ , ∗ , ∗ , ∗ , ∗ , ∗ , ∗ D d C , , , , C h C ∗ , ∗ , ∗ , ∗ , ∗ , ∗ , ∗ , , , ∗ , ∗ , ∗ D C , , , , D d C O C composite band rep-resentations for the double groups. The first column gives the maximal site-symmetry group, G q , which induces the compositerepresentation. Point groups symbols are given using Schoenflies notation (e.g C s is the point group generated by reflection).The second column gives the site-symmetry group, G q , into whose band representations this composite representation can bereduced. The third column gives the intersection group, G = G q ∩ G q . The fourth column gives the dimension of the irrepwhich induces the composite band rep. The fifth column indicates the space groups for which this occurs. An asterisk ( ∗ )indicates that while the band rep is disconnected in momentum space when time-reversal symmetry is ignored, there are extraconnectivity constraints imposed by Kramers’s theorem when TR is present. This can occur when the representation σ of G induces two one-dimensional representations of G q that are not momentum-space time reversal invariant in isolation.Site symmetry group ( G q ) Reducing group ( G q ) Space Group Number S C h , , , D , , , , , , , , , , D D d T , S . For the space groups listed in this table, this band representationdecomposes through G = C into a composite band representation induced from the reducing group G q . The first columngives the reducing group, while the second column gives the associated space groups for which the exception occurs. PG PGSymbol s p dC C i ¯1 Γ +1 − +1 C Γ ⊕ ⊕ C s m Γ ⊕ Γ ⊕ C h /m Γ +1 Γ − ⊕ − +1 ⊕ +2 D
222 Γ Γ ⊕ Γ ⊕ Γ ⊕ Γ ⊕ Γ ⊕ Γ C v mm Γ ⊕ Γ ⊕ Γ ⊕ Γ ⊕ Γ ⊕ Γ D h mmm Γ +1 Γ − ⊕ Γ − ⊕ Γ − +1 ⊕ Γ +2 ⊕ Γ +3 ⊕ Γ +4 C Γ ⊕ Γ ⊕ Γ Γ ⊕ ⊕ Γ ⊕ Γ S ¯4 Γ Γ ⊕ Γ ⊕ Γ Γ ⊕ ⊕ Γ ⊕ Γ C h /m Γ +1 Γ − ⊕ Γ − ⊕ Γ − Γ +1 ⊕ +2 ⊕ Γ +3 ⊕ Γ +4 D
422 Γ Γ ⊕ Γ Γ ⊕ Γ ⊕ Γ ⊕ Γ C v mm Γ Γ ⊕ Γ Γ ⊕ Γ ⊕ Γ ⊕ Γ D d ¯42 m Γ Γ ⊕ Γ Γ ⊕ Γ ⊕ Γ ⊕ Γ D h /mmm Γ +1 Γ − ⊕ Γ − Γ +1 ⊕ Γ +2 ⊕ Γ +4 ⊕ Γ +5 C ⊕ Γ ⊕ Γ Γ ⊕ ⊕ C i ¯3 Γ1 + Γ − ⊕ Γ − ⊕ Γ − Γ +1 ⊕ +2 ⊕ +3 D
32 Γ Γ ⊕ Γ Γ ⊕ C v m Γ Γ ⊕ Γ Γ ⊕ D d ¯3 m Γ +1 Γ − ⊕ Γ − Γ +1 ⊕ +3 C Γ ⊕ Γ ⊕ Γ Γ ⊕ Γ ⊕ Γ ⊕ Γ ⊕ Γ C h ¯6 Γ Γ ⊕ Γ ⊕ Γ Γ ⊕ Γ ⊕ Γ ⊕ Γ ⊕ Γ C h /m Γ +1 Γ − ⊕ Γ − ⊕ Γ − Γ +1 ⊕ Γ +3 ⊕ Γ +4 ⊕ Γ +5 ⊕ Γ +6 D
622 Γ Γ ⊕ Γ Γ ⊕ Γ ⊕ Γ C v mm Γ Γ ⊕ Γ Γ ⊕ Γ ⊕ Γ D h ¯62 m Γ Γ ⊕ Γ Γ ⊕ Γ ⊕ Γ D h /mmm Γ +1 Γ − ⊕ Γ − Γ +1 ⊕ Γ +5 ⊕ Γ +6 T
23 Γ Γ Γ ⊕ Γ ⊕ Γ T h m ¯3 Γ +1 Γ − Γ +2 ⊕ Γ +3 ⊕ Γ +4 O
432 Γ Γ Γ ⊕ Γ T d ¯43 m Γ Γ Γ ⊕ Γ O h m ¯3 m Γ +1 Γ − Γ +3 ⊕ Γ − TABLE S13. Decompositions of the representations spanned by spinless s, p and d orbitals into point group representations .The first column gives the point group symbol in Schoenflies notation, listed in the conventional order, and the second columngives point group symbol in Hermann-Mauguin notation. s orbitals transform in the point group representation listed inthe third column. p orbitals transform in the representation listed in the fourth column, and d orbitals transform in therepresentation listed in the last column. The representation labels correspond to the labelling of little group representations atthe Γ point; the notation matches the Bilbao Crystallographic Server . PG PGSymbol s p dC C i ¯1 2¯Γ C ⊕ ¯Γ ⊕ ⊕ C s m ¯Γ ⊕ ¯Γ ⊕ ⊕ C h /m ¯Γ ⊕ ¯Γ ⊕ ⊕ D
222 ¯Γ C v mm D h mmm ¯Γ C ⊕ ¯Γ ⊕ ⊕ ¯Γ ⊕ ¯Γ ⊕ ⊕ ⊕ S ¯4 ¯Γ ⊕ ¯Γ ¯Γ ⊕ ¯Γ ⊕ ⊕ ⊕ ⊕ ⊕ C h /m ¯Γ ⊕ ¯Γ ⊕ ⊕ ¯Γ ⊕ ¯Γ ⊕ ⊕ ⊕ D
422 ¯Γ ⊕ ¯Γ ⊕ C v mm ¯Γ ⊕ ¯Γ ⊕ D d ¯42 m ¯Γ ⊕ ¯Γ ⊕ D h /mmm ¯Γ ⊕ ¯Γ ⊕ C ⊕ ¯Γ ⊕ ⊕ ⊕ ⊕ C i ¯3 ¯Γ ⊕ ¯Γ ⊕ ⊕ ⊕ ⊕ D
32 ¯Γ ⊕ ¯Γ ⊕ ¯Γ ⊕ ⊕ C v m ¯Γ ⊕ ¯Γ ⊕ ¯Γ ⊕ ⊕ D d ¯3 m ¯Γ ⊕ ¯Γ ⊕ ¯Γ ⊕ ⊕ C ⊕ ¯Γ ⊕ ⊕ ¯Γ ⊕ ¯Γ ⊕ ⊕ ⊕ ⊕ ¯Γ ⊕ ¯Γ C h ¯6 ¯Γ ⊕ ¯Γ ⊕ ⊕ ¯Γ ⊕ ¯Γ ⊕ ⊕ ⊕ ⊕ ¯Γ ⊕ ¯Γ C h /m ¯Γ ⊕ ¯Γ ⊕ ⊕ ¯Γ ⊕ ¯Γ ⊕ ⊕ ⊕ ⊕ ¯Γ ⊕ ¯Γ D
622 ¯Γ ⊕ ¯Γ ⊕ ⊕ ¯Γ C v mm ¯Γ ⊕ ¯Γ ⊕ ⊕ ¯Γ D h ¯62 m ¯Γ ⊕ ¯Γ ⊕ ⊕ ¯Γ D h /mmm ¯Γ ⊕ ¯Γ ⊕ ⊕ ¯Γ T
23 ¯Γ ¯Γ ⊕ ¯Γ ⊕ ¯Γ ¯Γ ⊕ ⊕ T h m ¯3 ¯Γ ¯Γ ⊕ ¯Γ ⊕ ¯Γ ¯Γ ⊕ ⊕ O
432 ¯Γ ¯Γ ⊕ ¯Γ ⊕ ¯Γ T d ¯43 m ¯Γ ¯Γ ⊕ ¯Γ ⊕ ¯Γ O h m ¯3 m ¯Γ ¯Γ ⊕ ¯Γ ⊕ ¯Γ TABLE S14. Decompositions of the representations spanned by spinful s, p and d orbitals (assuming spin-1/2 electrons) intopoint group representations . The first column gives the point group symbol in Schoenflies notation, listed in the conventionalorder, and the second column gives point group symbol in Hermann-Mauguin notation. s orbitals transform according to thepoint group representation listed in the third column. p orbitals transform according to the representation listed in the fourthcolumn, and d orbitals transform according to the representation listed in the last column. The representation labels correspondto the labelling of little group representations at the Γ point; the notation matches the Bilbao Crystallographic Server . SG Mat. SG Mat. SG Mat. SG Mat. SG Mat.2 P ¯1 IrTe P Si R P
22 Ir Zr P m ¯3 m LaIn P Ge LaPt P bm La S P ¯3 NW P
22 Ge Ta 223
P m ¯3 n IrTi P /c AuCrTe P cc TaTe R ¯3 Ir Te P
22 Ni N 224
P n ¯3 n AgO P /c AgF Na I md LaPtSi 149 P
312 TiO P cm IrMg F m ¯3 m BiLa26
P mc
21 In LaPd P ¯42 m Na Sn 150 P
321 Li Pb P mc Au Sr F m ¯3 c NaZn P nn I ¯4 c ) P
21 Ga Ni P ¯6 m F d ¯3 m RbBi Cmc AsNi 122 I ¯4 d FeAgS R
32 Ni S P ¯6 c Ia ¯3 d Ga Ni Aem P /mmm InSePd P m AuCd 189 P ¯62 m GaAg F dd Y P /mnc CSc P c IrLi Si P ¯62 c HfSnRh52
P nna Bi Sr P /nmm LaTe R m As Sn P /mmm Ga La55
P bam Al Pt P /ncc Ge La R c Li ReO P /mcm Sr Sb P nnm
AlAu P /mmc La(BC) P ¯31 m Ag (PbO ) P /mmc Ge Li Zn59
P nmm Ag Sn 136 P /mnm ReO P ¯3 m F 198 P P bca
AgF P /mcm Ge La P ¯3 c CuPb P m ¯3 Au In Na P nma
AgSr 139 I /mmm LiTlPd R ¯3 m Zr Te P 205
P a ¯3 PdN Cmcm
BiZr 140 I /mcm Te Tl R ¯3 c Ir Mg Ia ¯3 Mg Bi Cmce Al Ge La I /amd NiTi P AlCaSi 212 P
32 BaSi Cmce Al Ge La I /acd IrSn P ¯6 Li Ni P
32 Ni W N74
Imma La Pd Si 143 P P /m Rb SnTe P
32 La SbI P /m AlNi Zr P IrGe P /m V S P ¯43 m Li Al Si TABLE S15. Excerpt of semimetal candidates, with electron filling smaller than the number of bands in the smallest PEBR.This criteria ensures that all materials shown are partially filled (semi-)metals with SOC. A complete list will be presented ina future work.
VII. TABLE OF EBRS AND PEBRS
Here we give the table of elementary and physically elementary band representations induced from the maximalWykoff positions in all 230 space groups in a condensed form. The column labeled “SG” gives the space group number.“MWP” gives the standard name of the maximal Wyckoff position, and “WM” gives its multiplicity in the primitivecell. “PG” is the point group number of for the site symmetry group, and “Irrep” gives the name of the site-symmetrygroup representation from which each band representation is induced. The reperesentations are labelled using thenotation of Stokes, Cordes, and Campbell . The column “Dim” denotes the dimension of the point stabilizer groupirrep. The column “KR” denotes whether the band representation is also a physical band representation. Thosewith a “1” in this column are PEBRs as is, Those with a “2” join with copies of themselves when TR symmetry isincluded. Finally, EBRs labelled by “ f ” (for first) pair with their conjugate BR labelled by “ s ” (and listed directlybelow) when TR symmetry is added. The column labelled “Bands” gives the total number of bands in the physicalband representation (to obtain the number of bands in the EBR without TR, divide this number by 1 if the entryin KR is 1, and 2 otherwise). The column “Re” indicates whether the given band representation can be made time-reversal invariant in momentum space: a 1 in this column indicates that TR symmetry is satisfied at each k point,while a 2 indicates that the given band representation must be connected in momentum space with its TR conjugate.In particular, those band representations induced from 1 d site-symmetry representations and with a 1 in the “Re”column are prime candidates for topological insulators, as discussed in Section IV. A of the main text. Finally, thecolumns “E” and “PE” indicate whether the given band representation is an exception (in the language of Sec. I andTables S10, S11, and S12), with and without TR symmetry respectively. An “e” in either of these columns indicateselementary, while a “c” indicates composite. This full set of data can be accessed in uncondensed form through theBANDREP program on the Bilbao Crystallographic Server . SG MWP WM PG Irrep Dim KR Bands Re E PE SG MWP WM PG Irrep Dim KR Bands Re E PE1 1 a e e
131 2 d − e e a e e
131 2 d +4 e e a +1 e e
131 2 d − e e a − e e
131 2 d +3 e e a e e
131 2 d − e e a e e
131 2 d e e b +1 e e
131 2 d e e b − e e
131 2 e e e b e e
131 2 e e e b e e
131 2 e e e c +1 e e
131 2 e e e c − e e
131 2 e c c c e e
131 2 e e e c e e
131 2 e e e d +1 e e
131 2 f e e d − e e
131 2 f e e d e e
131 2 f e e d e e
131 2 f e e e +1 e e
131 2 f c c e − e e
131 2 f e e e e e
131 2 f e e e e e
132 2 a +1 e e f +1 e e
132 2 a − e e f − e e
132 2 a +2 e e f e e
132 2 a − e e f e e
132 2 a +4 e e g +1 e e
132 2 a − e e g − e e
132 2 a +3 e e g e e
132 2 a − e e g e e
132 2 a e e h +1 e e
132 2 a e e h − e e
132 2 b e e h e e
132 2 b e e h e e
132 2 b e e a e e
132 2 b e e a e e
132 2 b c c a f e e
132 2 b e e a s e
132 2 b e e b e e
132 2 c +1 e e b e e
132 2 c − e e b f e e
132 2 c +2 e e b s e
132 2 c − e e c e e
132 2 c +4 e e c e e
132 2 c − e e c f e e
132 2 c +3 e e c s e
132 2 c − e e d e e
132 2 c e e d e e
132 2 c e e d f e e
132 2 d e e d s e
132 2 d e e a e e
132 2 d e e a e e
132 2 d e e a e e
132 2 d c c a e e
132 2 d e e a f e e
132 2 d e e a s e
132 4 e e e b e e
132 4 e e e b e e
132 4 e e e b f e e
132 4 e e e b s e
132 4 e c e a e e
132 4 f +1 e e a e e
132 4 f − e e a f e e
132 4 f +2 e e a s e
132 4 f − e e b e e
132 4 f f e e b e e
132 4 f s e b f e e
132 4 f f e e b s e
132 4 f s e a e e
133 4 a e e a e e
133 4 a e e a e e
133 4 a e e a e e
133 4 a e e a f e e
133 4 a e e a s e
133 4 b e e a e e
133 4 b e e a e e
133 4 b e e
10 1 a +1 e e
133 4 b e e
10 1 a − e e
133 4 b c e
10 1 a +2 e e
133 4 c e e
10 1 a − e e
133 4 c e e
10 1 a f e e
133 4 c e e
10 1 a s e
133 4 c e e
10 1 a f e e
133 4 c e e
10 1 a s e
133 4 d e e
10 1 b +1 e e
133 4 d e e
10 1 b − e e
133 4 d f e c
10 1 b +2 e e
133 4 d s e
10 1 b − e e
133 4 d f e e
10 1 b f e e
133 4 d s e
10 1 b s e
133 4 d f e e
10 1 b f e e
133 4 d s e
10 1 b s e
133 8 e +1 e e
10 1 c +1 e e
133 8 e − e e
10 1 c − e e
133 8 e e e
10 1 c +2 e e
133 8 e e e
10 1 c − e e
134 2 a e e
10 1 c f e e
134 2 a e e
10 1 c s e
134 2 a e e
10 1 c f e e
134 2 a e e
10 1 c s e
134 2 a e e
10 1 d +1 e e
134 2 a e e
10 1 d − e e
134 2 a e e
10 1 d +2 e e
134 2 b e e
10 1 d − e e
134 2 b e e
10 1 d f e e
134 2 b e e
10 1 d s e
134 2 b e e
10 1 d f e e
134 2 b e e
10 1 d s e
134 2 b e e
10 1 e +1 e e
134 2 b e e
10 1 e − e e
134 4 c e e
10 1 e +2 e e
134 4 c e e
10 1 e − e e
134 4 c e e
10 1 e f e e
134 4 c e e
10 1 e s e
134 4 c c e
10 1 e f e e
134 4 d e e
10 1 e s e
134 4 d e e
10 1 f +1 e e
134 4 d e e
10 1 f − e e
134 4 d e e
10 1 f +2 e e
134 4 d c e
10 1 f − e e
134 4 e +1 e e
10 1 f f e e
134 4 e − e e
10 1 f s e
134 4 e +2 e e
10 1 f f e e
134 4 e − e e
10 1 f s e
134 4 e f e e
10 1 g +1 e e
134 4 e s e
10 1 g − e e
134 4 e f e e
10 1 g +2 e e
134 4 e s e
10 1 g − e e
134 4 f +1 e e
10 1 g f e e
134 4 f − e e
10 1 g s e
134 4 f +2 e e
10 1 g f e e
134 4 f − e e
10 1 g s e
134 4 f f e e
10 1 h +1 e e
134 4 f s e
10 1 h − e e
134 4 f f e e
10 1 h +2 e e
134 4 f s e
10 1 h − e e
135 4 a +1 e e
10 1 h f e e
135 4 a − e e
10 1 h s e
135 4 a +2 e e
10 1 h f e e
135 4 a − e e
10 1 h s e
135 4 a f e e
11 2 a +1 e e
135 4 a s e
11 2 a − e e
135 4 a f e e
11 2 a e e
135 4 a s e
11 2 a e e
135 4 b e e
11 2 b +1 e e
135 4 b e e
11 2 b − e e
135 4 b f e c
11 2 b e e
135 4 b s e
11 2 b e e
135 4 b f e e
11 2 c +1 e e
135 4 b s e
11 2 c − e e
135 4 b f e e
11 2 c e e
135 4 b s e
11 2 c e e
135 4 c +1 e e
11 2 d +1 e e
135 4 c − e e
11 2 d − e e
135 4 c +2 e e
11 2 d e e
135 4 c − e e
11 2 d e e
135 4 c f e e
11 2 e e e
135 4 c s e
11 2 e e e
135 4 c f e e
11 2 e f e e
135 4 c s e
11 2 e s e
135 4 d e e
12 2 a +1 e e
135 4 d e e
12 2 a − e e
135 4 d e e
12 2 a +2 e e
135 4 d e e
12 2 a − e e
135 4 d c e
12 2 a f e e
136 2 a +1 e e
12 2 a s e
136 2 a − e e
12 2 a f e e
136 2 a +2 e e
12 2 a s e
136 2 a − e e
12 2 b +1 e e
136 2 a +4 e e
12 2 b − e e
136 2 a − e e
12 2 b +2 e e
136 2 a +3 e e
12 2 b − e e
136 2 a − e e
12 2 b f e e
136 2 a e e
12 2 b s e
136 2 a e e
12 2 b f e e
136 2 b +1 e e
12 2 b s e
136 2 b − e e
12 2 c +1 e e
136 2 b +2 e e
12 2 c − e e
136 2 b − e e
12 2 c +2 e e
136 2 b +4 e e
12 2 c − e e
136 2 b − e e
12 2 c f e e
136 2 b +3 e e
12 2 c s e
136 2 b − e e
12 2 c f e e
136 2 b e e
12 2 c s e
136 2 b e e
12 2 d +1 e e
136 4 c +1 e e
12 2 d − e e
136 4 c − e e
12 2 d +2 e e
136 4 c +2 e e
12 2 d − e e
136 4 c − e e
12 2 d f e e
136 4 c f e e
12 2 d s e
136 4 c s e
12 2 d f e e
136 4 c f e e
12 2 d s e
136 4 c s e
12 4 e +1 e e
136 4 d e e
12 4 e − e e
136 4 d e e
12 4 e e e
136 4 d f e c
12 4 e e e
136 4 d s e
12 4 f +1 e e
136 4 d f e e
12 4 f − e e
136 4 d s e
12 4 f e e
136 4 d f e e
12 4 f e e
136 4 d s e
13 2 a +1 e e
137 2 a e e
13 2 a − e e
137 2 a e e
13 2 a e e
137 2 a e e
13 2 a e e
137 2 a e e
13 2 b +1 e e
137 2 a e e
13 2 b − e e
137 2 a e e
13 2 b e e
137 2 a e e
13 2 b e e
137 2 b e e
13 2 c +1 e e
137 2 b e e
13 2 c − e e
137 2 b e e
13 2 c e e
137 2 b e e
13 2 c e e
137 2 b e e
13 2 d +1 e e
137 2 b e e
13 2 d − e e
137 2 b e e
13 2 d e e
137 4 d e e
13 2 d e e
137 4 d e e
13 2 e e e
137 4 d e e
13 2 e e e
137 4 d e e
13 2 e f e e
137 4 d c e
13 2 e s e
137 8 e +1 e e
13 2 f e e
137 8 e − e e
13 2 f e e
137 8 e e e
13 2 f f e e
137 8 e e e
13 2 f s e
138 4 a e e
14 2 a +1 e e
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14 2 a − e e
138 4 a e e
14 2 a e e
138 4 a e e
14 2 a e e
138 4 a c e
14 2 b +1 e e
138 4 b e e
14 2 b − e e
138 4 b e e
14 2 b e e
138 4 b f e c
14 2 b e e
138 4 b s e
14 2 c +1 e e
138 4 b f e e
14 2 c − e e
138 4 b s e
14 2 c e e
138 4 b f e e
14 2 c e e
138 4 b s e
14 2 d +1 e e
138 4 c +1 e e
14 2 d − e e
138 4 c − e e
14 2 d e e
138 4 c +2 e e
14 2 d e e
138 4 c − e e
15 4 a +1 e e
138 4 c f e e
15 4 a − e e
138 4 c s e
15 4 a e e
138 4 c f e e
15 4 a e e
138 4 c s e
15 4 b +1 e e
138 4 d +1 e e
15 4 b − e e
138 4 d − e e
15 4 b e e
138 4 d +2 e e
15 4 b e e
138 4 d − e e
15 4 c +1 e e
138 4 d f e e
15 4 c − e e
138 4 d s e
15 4 c e e
138 4 d f e e
15 4 c e e
138 4 d s e
15 4 d +1 e e
138 4 e e e
15 4 d − e e
138 4 e e e
15 4 d e e
138 4 e e e
15 4 d e e
138 4 e e e
15 4 e e e
138 4 e c e
15 4 e e e
139 2 a +1 e e
15 4 e f e e
139 2 a − e e
15 4 e s e
139 2 a +3 e e
16 1 a e e
139 2 a − e e
16 1 a e e
139 2 a +2 e e
16 1 a e e
139 2 a − e e
16 1 a e e
139 2 a +4 e e
16 1 a e e
139 2 a − e e
16 1 b e e
139 2 a e e
16 1 b e e
139 2 a e e
16 1 b e e
139 2 a e e
16 1 b e e
139 2 a e e
16 1 b e e
139 2 a +5 e e
16 1 c e e
139 2 a − e e
16 1 c e e
139 2 b +1 e e
16 1 c e e
139 2 b − e e
16 1 c e e
139 2 b +3 e e
16 1 c e e
139 2 b − e e
16 1 d e e
139 2 b +2 e e
16 1 d e e
139 2 b − e e
16 1 d e e
139 2 b +4 e e
16 1 d e e
139 2 b − e e
16 1 d e e
139 2 b e e
16 1 e e e
139 2 b e e
16 1 e e e
139 2 b e e
16 1 e e e
139 2 b e e
16 1 e e e
139 2 b +5 e e
16 1 e e e
139 2 b − e e
16 1 f e e
139 4 c +1 e e
16 1 f e e
139 4 c − e e
16 1 f e e
139 4 c +2 e e
16 1 f e e
139 4 c − e e
16 1 f e e
139 4 c +4 e e
16 1 g e e
139 4 c − e e
16 1 g e e
139 4 c +3 e e
16 1 g e e
139 4 c − e e
16 1 g e e
139 4 c e e
16 1 g e e
139 4 c e e
16 1 h e e
139 4 d e e
16 1 h e e
139 4 d e e
16 1 h e e
139 4 d e e
16 1 h e e
139 4 d e e
16 1 h e e
139 4 d c c
17 2 a e e
139 4 d e e
17 2 a e e
139 4 d e e
17 2 a f e e
139 8 f +1 e e
17 2 a s e
139 8 f − e e
17 2 b e e
139 8 f +2 e e
17 2 b e e
139 8 f − e e
17 2 b f e e
139 8 f f e e
17 2 b s e
139 8 f s e
17 2 c e e
139 8 f f e e
17 2 c e e
139 8 f s e
17 2 c f e e
140 4 a e e
17 2 c s e
140 4 a e e
17 2 d e e
140 4 a e e
17 2 d e e
140 4 a e e
17 2 d f e e
140 4 a c e
17 2 d s e
140 4 a c e
18 2 a e e
140 4 a c e
18 2 a e e
140 4 b e e
18 2 a f e e
140 4 b e e
18 2 a s e
140 4 b e e
18 2 b e e
140 4 b e e
18 2 b e e
140 4 b c c
18 2 b f e e
140 4 b e e
18 2 b s e
140 4 b e e
19 4 a e e
140 4 c +1 e e
19 4 a e e
140 4 c − e e
20 4 a e e
140 4 c +2 e e
20 4 a e e
140 4 c − e e
20 4 a f e e
140 4 c f e e
20 4 a s e
140 4 c s e
20 4 b e e
140 4 c f e e
20 4 b e e
140 4 c s e
20 4 b f e e
140 4 c f e e
20 4 b s e
140 4 c s e
21 2 a e e
140 4 c f e e
21 2 a e e
140 4 c s e
21 2 a e e
140 4 c +4 f e e
21 2 a e e
140 4 c +3 s e
21 2 a e e
140 4 c − f e e
21 2 b e e
140 4 c − s e
21 2 b e e
140 4 d +1 e e
21 2 b e e
140 4 d − e e
21 2 b e e
140 4 d +2 e e
21 2 b e e
140 4 d − e e
21 2 c e e
140 4 d +4 e e
21 2 c e e
140 4 d − e e
21 2 c e e
140 4 d +3 e e
21 2 c e e
140 4 d − e e
21 2 c e e
140 4 d e e
21 2 d e e
140 4 d e e
21 2 d e e
140 8 e +1 e e
21 2 d e e
140 8 e − e e
21 2 d e e
140 8 e +2 e e
21 2 d e e
140 8 e − e e
21 4 k e e
140 8 e f e e
21 4 k e e
140 8 e s e
21 4 k f e e
140 8 e f e e
21 4 k s e
140 8 e s e
22 4 a e e
141 4 a e e
22 4 a e e
141 4 a e e
22 4 a e e
141 4 a e e
22 4 a e e
141 4 a e e
22 4 a e e
141 4 a e e
22 4 b e e
141 4 a e e
22 4 b e e
141 4 a e e
22 4 b e e
141 4 b e e
22 4 b e e
141 4 b e e
22 4 b e e
141 4 b e e
22 4 c e e
141 4 b e e
22 4 c e e
141 4 b e e
22 4 c e e
141 4 b e e
22 4 c e e
141 4 b e e
22 4 c e e
141 8 c +1 e e
22 4 d e e
141 8 c − e e
22 4 d e e
141 8 c +2 e e
22 4 d e e
141 8 c − e e
22 4 d e e
141 8 c f e e
22 4 d e e
141 8 c s e
23 2 a e e
141 8 c f e e
23 2 a e e
141 8 c s e
23 2 a e e
141 8 d +1 e e
23 2 a e e
141 8 d − e e
23 2 a e e
141 8 d +2 e e
23 2 b e e
141 8 d − e e
23 2 b e e
141 8 d f e e
23 2 b e e
141 8 d s e
23 2 b e e
141 8 d f e e
23 2 b e e
141 8 d s e
23 2 c e e
142 8 a e e
23 2 c e e
142 8 a e e
23 2 c e e
142 8 a f e c
23 2 c e e
142 8 a s e
23 2 c e e
142 8 a f e e
23 2 d e e
142 8 a s e
23 2 d e e
142 8 a f e e
23 2 d e e
142 8 a s e
23 2 d e e
142 8 b e e
23 2 d e e
142 8 b e e
24 4 a e e
142 8 b e e
24 4 a e e
142 8 b e e
24 4 a f e e
142 8 b c e
24 4 a s e
142 16 c +1 e e
24 4 b e e
142 16 c − e e
24 4 b e e
142 16 c e e
24 4 b f e e
142 16 c e e
24 4 b s e
142 16 e e e
24 4 c e e
142 16 e e e
24 4 c e e
142 16 e f
16 1 e e
24 4 c f e e
142 16 e s e
24 4 c s e
143 1 a e e
25 1 a e e
143 1 a e e
25 1 a e e
143 1 a f e e
25 1 a e e
143 1 a s e
25 1 a e e
143 1 a f e e
25 1 a e e
143 1 a s e
25 1 b e e
143 1 b e e
25 1 b e e
143 1 b e e
25 1 b e e
143 1 b f e e
25 1 b e e
143 1 b s e
25 1 b e e
143 1 b f e e
25 1 c e e
143 1 b s e
25 1 c e e
143 1 c e e
25 1 c e e
143 1 c e e
25 1 c e e
143 1 c f e e
25 1 c e e
143 1 c s e
25 1 d e e
143 1 c f e e
25 1 d e e
143 1 c s e
25 1 d e e
144 3 a e e
25 1 d e e
144 3 a e e
25 1 d e e
145 3 a e e
26 2 a e e
145 3 a e e
26 2 a e e
146 3 a e e
26 2 a f e e
146 3 a e e
26 2 a s e
146 3 a f e e
26 2 b e e
146 3 a s e
26 2 b e e
146 3 a f e e
26 2 b f e e
146 3 a s e
26 2 b s e
147 1 a +1 e e
27 2 a e e
147 1 a − e e
27 2 a e e
147 1 a e e
27 2 a f e e
147 1 a e e
27 2 a s e
147 1 a f e e
27 2 b e e
147 1 a s e
27 2 b e e
147 1 a +3 f e e
27 2 b f e e
147 1 a +2 s e
27 2 b s e
147 1 a f e e
27 2 c e e
147 1 a s e
27 2 c e e
147 1 a − f e e
27 2 c f e e
147 1 a − s e
27 2 c s e
147 1 b +1 e e
27 2 d e e
147 1 b − e e
27 2 d e e
147 1 b e e
27 2 d f e e
147 1 b e e
27 2 d s e
147 1 b f e e
28 2 a e e
147 1 b s e
28 2 a e e
147 1 b +3 f e e
28 2 a f e e
147 1 b +2 s e
28 2 a s e
147 1 b f e e
28 2 b e e
147 1 b s e
28 2 b e e
147 1 b − f e e
28 2 b f e e
147 1 b − s e
28 2 b s e
147 2 d e e
28 2 c e e
147 2 d e e
28 2 c e e
147 2 d f e e
28 2 c f e e
147 2 d s e
28 2 c s e
147 2 d f e e
29 4 a e e
147 2 d s e
29 4 a e e
147 3 e +1 e e
30 2 a e e
147 3 e − e e
30 2 a e e
147 3 e e e
30 2 a f e e
147 3 e e e
30 2 a s e
147 3 f +1 e e
30 2 b e e
147 3 f − e e
30 2 b e e
147 3 f e e
30 2 b f e e
147 3 f e e
30 2 b s e
148 3 a +1 e e
31 2 a e e
148 3 a − e e
31 2 a e e
148 3 a e e
31 2 a f e e
148 3 a e e
31 2 a s e
148 3 a f e e
32 2 a e e
148 3 a s e
32 2 a e e
148 3 a +3 f e e
32 2 a f e e
148 3 a +2 s e
32 2 a s e
148 3 a f e e
32 2 b e e
148 3 a s e
32 2 b e e
148 3 a − f e e
32 2 b f e e
148 3 a − s e
32 2 b s e
148 3 b +1 e e
33 4 a e e
148 3 b − e e
33 4 a e e
148 3 b e e
34 2 a e e
148 3 b e e
34 2 a e e
148 3 b f e e
34 2 a f e e
148 3 b s e
34 2 a s e
148 3 b +3 f e e
34 2 b e e
148 3 b +2 s e
34 2 b e e
148 3 b f e e
34 2 b f e e
148 3 b s e
34 2 b s e
148 3 b − f e e
35 2 a e e
148 3 b − s e
35 2 a e e
148 9 d +1 e e
35 2 a e e
148 9 d − e e
35 2 a e e
148 9 d e e
35 2 a e e
148 9 d e e
35 2 b e e
148 9 e +1 e e
35 2 b e e
148 9 e − e e
35 2 b e e
148 9 e e e
35 2 b e e
148 9 e e e
35 2 b e e
149 1 a e e
35 4 c e e
149 1 a e e
35 4 c e e
149 1 a e e
35 4 c f e e
149 1 a e e
35 4 c s e
149 1 a f e e
36 4 a e e
149 1 a s e
36 4 a e e
149 1 b e e
36 4 a f e e
149 1 b e e
36 4 a s e
149 1 b e e
37 4 a e e
149 1 b e e
37 4 a e e
149 1 b f e e
37 4 a f e e
149 1 b s e
37 4 a s e
149 1 c e e
37 4 b e e
149 1 c e e
37 4 b e e
149 1 c e e
37 4 b f e e
149 1 c e e
37 4 b s e
149 1 c f e e
37 4 c e e
149 1 c s e
37 4 c e e
149 1 d e e
37 4 c f e e
149 1 d e e
37 4 c s e
149 1 d e e
38 2 a e e
149 1 d e e
38 2 a e e
149 1 d f e e
38 2 a e e
149 1 d s e
38 2 a e e
149 1 e e e
38 2 a e e
149 1 e e e
38 2 b e e
149 1 e e e
38 2 b e e
149 1 e e e
38 2 b e e
149 1 e f e e
38 2 b e e
149 1 e s e
38 2 b e e
149 1 f e e
39 4 a e e
149 1 f e e
39 4 a e e
149 1 f e e
39 4 a f e e
149 1 f e e
39 4 a s e
149 1 f f e e
39 4 b e e
149 1 f s e
39 4 b e e
150 1 a e e
39 4 b f e e
150 1 a e e
39 4 b s e
150 1 a e e
39 4 c e e
150 1 a e e
39 4 c e e
150 1 a f e e
39 4 c f e e
150 1 a s e
39 4 c s e
150 1 b e e
40 4 a e e
150 1 b e e
40 4 a e e
150 1 b e e
40 4 a f e e
150 1 b e e
40 4 a s e
150 1 b f e e
40 4 b e e
150 1 b s e
40 4 b e e
150 2 d e e
40 4 b f e e
150 2 d e e
40 4 b s e
150 2 d f e e
41 4 a e e
150 2 d s e
41 4 a e e
150 2 d f e e
41 4 a f e e
150 2 d s e
41 4 a s e
151 3 a e e
42 4 a e e
151 3 a e e
42 4 a e e
151 3 a f e e
42 4 a e e
151 3 a s e
42 4 a e e
151 3 b e e
42 4 a e e
151 3 b e e
42 8 b e e
151 3 b f e e
42 8 b e e
151 3 b s e
42 8 b f e e
152 3 a e e
42 8 b s e
152 3 a e e
43 8 a e e
152 3 a f e e
43 8 a e e
152 3 a s e
43 8 a f e e
152 3 b e e
43 8 a s e
152 3 b e e
44 2 a e e
152 3 b f e e
44 2 a e e
152 3 b s e
44 2 a e e
153 3 a e e
44 2 a e e
153 3 a e e
44 2 a e e
153 3 a f e e
44 2 b e e
153 3 a s e
44 2 b e e
153 3 b e e
44 2 b e e
153 3 b e e
44 2 b e e
153 3 b f e e
44 2 b e e
153 3 b s e
45 4 a e e
154 3 a e e
45 4 a e e
154 3 a e e
45 4 a f e e
154 3 a f e e
45 4 a s e
154 3 a s e
45 4 b e e
154 3 b e e
45 4 b e e
154 3 b e e
45 4 b f e e
154 3 b f e e
45 4 b s e
154 3 b s e
46 4 a e e
155 3 a e e
46 4 a e e
155 3 a e e
46 4 a f e e
155 3 a e e
46 4 a s e
155 3 a e e
46 4 b e e
155 3 a f e e
46 4 b e e
155 3 a s e
46 4 b f e e
155 3 b e e
46 4 b s e
155 3 b e e
47 1 a +1 e e
155 3 b e e
47 1 a − e e
155 3 b e e
47 1 a +2 e e
155 3 b f e e
47 1 a − e e
155 3 b s e
47 1 a +4 e e
156 1 a e e
47 1 a − e e
156 1 a e e
47 1 a +3 e e
156 1 a e e
47 1 a − e e
156 1 a e e
47 1 a e e
156 1 a f e e
47 1 a e e
156 1 a s e
47 1 b +1 e e
156 1 b e e
47 1 b − e e
156 1 b e e
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175 2 c s e
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64 8 c e e
175 2 d f e e
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175 2 d f e e
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175 2 d s e
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65 2 a − e e
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65 2 a +4 e e
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65 2 a +3 e e
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65 2 b − e e
175 3 g f e e
65 2 b +2 e e
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65 2 b − e e
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65 2 c +1 e e
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65 2 c +3 e e
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176 2 b − e e
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227 16 c +2 e e
128 2 a f e e
227 16 c − e e
128 2 a s e
227 16 c e e
128 2 a f e e
227 16 c e e
128 2 a s e
227 16 c +3 e e
128 2 a f e e
227 16 c − e e
128 2 a s e
227 16 c f e e
128 2 a +4 f e e
227 16 c s e
128 2 a +3 s e
227 16 c f e e
128 2 a − f e e
227 16 c s e
128 2 a − s e
227 16 d +1 e e
128 2 b +1 e e
227 16 d − e e
128 2 b − e e
227 16 d +2 e e
128 2 b +2 e e
227 16 d − e e
128 2 b − e e
227 16 d e e
128 2 b f e e
227 16 d e e
128 2 b s e
227 16 d +3 e e
128 2 b f e e
227 16 d − e e
128 2 b s e
227 16 d f e e
128 2 b f e e
227 16 d s e
128 2 b s e
227 16 d f e e
128 2 b f e e
227 16 d s e
128 2 b s e
228 16 a e e
128 2 b +4 f e e
228 16 a e e
128 2 b +3 s e
228 16 a e e
128 2 b − f e e
228 16 a f e e
128 2 b − s e
228 16 a s e
128 4 c +1 e e
228 16 a f
16 1 e e
128 4 c − e e
228 16 a s e
128 4 c +2 e e
228 32 b e e
128 4 c − e e
228 32 b e e
128 4 c f e e
228 32 b c e
128 4 c s e
228 32 b c e
128 4 c f e e
228 32 b f
16 1 e e
128 4 c s e
228 32 b s e
128 4 d e e
228 32 c +1 e e
128 4 d e e
228 32 c − e e
128 4 d e e
228 32 c e e
128 4 d e e
228 32 c e e
128 4 d c e
228 32 c f
16 2 e e
129 2 a e e
228 32 c s e
129 2 a e e
228 32 c +3 f
16 2 e e
129 2 a e e
228 32 c +2 s e
129 2 a e e
228 32 c f
16 2 e e
129 2 a e e
228 32 c s e
129 2 a e e
228 32 c − f
16 2 e e
129 2 a e e
228 32 c − s e
129 2 b e e
228 48 d
12 10 Γ e e
129 2 b e e
228 48 d
12 10 Γ e e
129 2 b e e
228 48 d
12 10 Γ f
24 2 e c
129 2 b e e
228 48 d
12 10 Γ s e
129 2 b e e
228 48 d
12 10 ¯Γ f
24 2 e e
129 2 b e e
228 48 d
12 10 ¯Γ s e
129 2 b e e
228 48 d
12 10 ¯Γ f
24 2 e e
129 2 c e e
228 48 d
12 10 ¯Γ s e
129 2 c e e
229 2 a +1 e e
129 2 c e e
229 2 a − e e
129 2 c e e
229 2 a +2 e e
129 2 c e e
229 2 a − e e
129 2 c e e
229 2 a e e
129 2 c e e
229 2 a e e
129 4 d +1 e e
229 2 a e e
129 4 d − e e
229 2 a e e
129 4 d +2 e e
229 2 a +3 e e
129 4 d − e e
229 2 a − e e
129 4 d f e e
229 2 a e e
129 4 d s e
229 2 a e e
129 4 d f e e
229 2 a +4 e e
129 4 d s e
229 2 a − e e
129 4 e +1 e e
229 2 a +5 e e
129 4 e − e e
229 2 a − e e
129 4 e +2 e e
229 6 b +1 e e
129 4 e − e e
229 6 b − e e
129 4 e f e e
229 6 b +3 e e
129 4 e s e
229 6 b − e e
129 4 e f e e
229 6 b +2 e e
129 4 e s e
229 6 b − e e
130 4 a e e
229 6 b +4 e e
130 4 a e e
229 6 b − e e
130 4 a e e
229 6 b e e
130 4 a e e
229 6 b e e
130 4 a c e
229 6 b e e
130 4 b e e
229 6 b e e
130 4 b e e
229 6 b +5 e e
130 4 b f e c
229 6 b − e e
130 4 b s e
229 8 c +1 e e
130 4 b f e e
229 8 c − e e
130 4 b s e
229 8 c +2 e e
130 4 b f e e
229 8 c − e e
130 4 b s e
229 8 c e e
130 4 c e e
229 8 c e e
130 4 c e e
229 8 c +3 e e
130 4 c f e e
229 8 c − e e
130 4 c s e
229 8 c f e e
130 4 c f e e
229 8 c s e
130 4 c s e
229 8 c f e e
130 4 c f e e
229 8 c s e
130 4 c s e
229 12 d e e
130 8 d +1 e e
229 12 d e e
130 8 d − e e
229 12 d e e
130 8 d e e
229 12 d e e
130 8 d e e
229 12 d c c
131 2 a +1 e e
229 12 d e e
131 2 a − e e
229 12 d e e
131 2 a +2 e e
230 16 a +1 e e
131 2 a − e e
230 16 a − e e
131 2 a +4 e e
230 16 a e e
131 2 a − e e
230 16 a e e
131 2 a +3 e e
230 16 a f
16 2 e e
131 2 a − e e
230 16 a s e
131 2 a e e
230 16 a +3 f
16 2 e e
131 2 a e e
230 16 a +2 s e
131 2 b +1 e e
230 16 a f
16 2 e e
131 2 b − e e
230 16 a s e
131 2 b +2 e e
230 16 a − f
16 2 e e
131 2 b − e e
230 16 a − s e
131 2 b +4 e e
230 16 b e e
131 2 b − e e
230 16 b e e
131 2 b +3 e e
230 16 b c e
131 2 b − e e
230 16 b c e
131 2 b e e
230 16 b f
16 1 e e
131 2 b e e
230 16 b s e
131 2 c +1 e e
230 24 c
12 6 Γ e e
131 2 c − e e
230 24 c
12 6 Γ e e
131 2 c +2 e e
230 24 c
12 6 Γ e e
131 2 c − e e
230 24 c
12 6 Γ e e
131 2 c +4 e e
230 24 c
12 6 ¯Γ c e
131 2 c − e e
230 24 d
12 10 Γ e e
131 2 c +3 e e
230 24 d
12 10 Γ e e
131 2 c − e e
230 24 d
12 10 Γ f
24 2 e c
131 2 c e e
230 24 d
12 10 Γ s e
131 2 c e e
230 24 d
12 10 ¯Γ f
24 2 e e
131 2 d +1 e e
230 24 d
12 10 ¯Γ s e
131 2 d − e e
230 24 d
12 10 ¯Γ f
24 2 e e
131 2 d +2 e e
230 24 d
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