Topological quantum chemistry

TL;DR

Topological Quantum Chemistry introduces a comprehensive band theory linking topology with chemical bonding, enabling classification and prediction of topological insulators across all crystal symmetries, addressing the scarcity of known TIs.

Contribution

It presents a complete electronic band theory that unifies topology and local chemical bonding, classifies all possible band structures, and predicts new topological insulators.

Findings

Classified all band structures for 230 crystal symmetries.

Identified which structures are topologically nontrivial.

Predicted numerous new topological insulators.

Abstract

The past decade's apparent success in predicting and experimentally discovering distinct classes of topological insulators (TIs) and semimetals masks a fundamental shortcoming: out of 200,000 stoichiometric compounds extant in material databases, only several hundred of them are topologically nontrivial. Are TIs that esoteric, or does this reflect a fundamental problem with the current piecemeal approach to finding them? To address this, we propose a new and complete electronic band theory that highlights the link between topology and local chemical bonding, and combines this with the conventional band theory of electrons. Topological Quantum Chemistry is a description of the universal global properties of all possible band structures and materials, comprised of a graph theoretical description of momentum space and a dual group theoretical description in real space. We classify the possible band structures for all 230 crystal symmetry groups that arise from local atomic orbitals, and show which are topologically nontrivial. We show how our topological band theory sheds new light on known TIs, and demonstrate the power of our method to predict a plethora of new TIs.

Figures

Tables

Table 1. Table 1: Summary of topological phase transitions between a system without SOC to a TI with SOC, for BRs induced from up to two irreps of the site-symmetry group of occupied WPs . When there is a single EBR in the first column, the system is necessarily a semimetal without SOC – these materials become TIs for arbitrarily small SOC. The first column gives the number of PEBRs directly straddling the Fermi level without SOC (per spin). The second column shows how this number of band representations changes when SOC is turned on.

Number of PEBRs without SOC	Number of PEBRs with SOC	Examples
1	1	Graphene, Bismuthene, Cu3SbS4
1	2	Bi1- square nets, Cu2SnHgSe4
2	2	Bi2Se3, KHgSb

Equations

$$\rho \downarrow G_{{\mathbf{k}}_t}\approx \underset{i}{⨁}{\tau }_i.$$

$$\sigma \downarrow G_{{\mathbf{k}}_t}\approx \underset{i}{⨁}{\tau }_i.$$

$${\rho }_{iG}={\rho }_i\uparrow G.$$

$$\eta \uparrow G_a\approx {\sigma }_v\oplus {\sigma }_c,\eta \uparrow G_m\approx {\rho }_v\oplus {\rho }_m.$$

Full text

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-range interactions, 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 understood in 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 400 materials out of 200,000 existent in crystal structure databases!), one may ask whether topological materials are truly so 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. The duality between real and momentum (direct and reciprocal) space suggests that properties of bands which are nonlocal in momentum space will manifest locally in real space. In this paper, we unify the real and momentum space descriptions of solids, and in doing so provide a new, powerful, complete and predictive band theory. Our procedure provides a complete understanding of the structure of bands in a material and links its topological to the chemical orbitals 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 several ingredients. First, we compile all the possible ways energy bands in a solid can be connected throughout the BZ to obtain all realizable band structures in all non-magnetic space-groups. Crystal symmetries place strong constraints on the allowed connections of bands. At high symmetry points, $\mathbf{k}_{i}$, in the BZ, Bloch functions are classified by irreducible representations (irreps) of the symmetry group of $\mathbf{k}_{i}$, which also determine the degeneracy. Away from these high symmetry points, fewer symmetry constraints exist, and some degeneracies are lowered. This is the heart of the $\mathbf{k}\cdot\mathbf{p}$ approachKittel (1987) to band structure, which gives a good description nearby high symmetry $\mathbf{k}-$points. However, the global band structure requires patching together the different $\mathbf{k}\cdot\mathbf{p}$ theories at various high symmetry points. Group theory places constraints – “compatibility relations” – on how this can be done. Each solution to these compatibility relations gives 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 in band theory to the graph-theoretic problem of constructing multipartite graphs. Classifying the allowed connectivities of energy bands becomes a combinatorial problem of graph enumeration: we present a fully tractable, algorithmic solution.

Second, we develop the tools to compute how the real-space orbitals in a material determine the symmetry character of the electronic bands. Given only the Wyckoff positions and the orbital symmetry ($s,p,d$) of the elements/orbitals in a material, we derive the symmetry character of all energy bands at all points in the BZ. We do this by extending the notion of a band representation (BR), first introduced in Refs. Zak, 1982; Bacry et al., 1988, to the physically relevant case of materials with spin-orbit coupling (SOC) and/or TR symmetry. A BR consists of all bands linked to localized orbitals respecting the crystal symmetry (and possibly TR). The set of BRs is strictly smaller than the set of groups of bands obtained from our graph theoryBacry (1993). We identify a special subset of “elementary” BRs (EBRs)Zak (1982); Evarestov and Smirnov (1997), elaborated upon in the Supplementary Material (SM), which are the smallest sets of bands derived from local atomic-like Wannier functions.Marzari et al. (2012) We work out all the (10300) different EBRs for all the SGs, Wyckoff positions, and orbitals, which we will present in a separate data paper.Elcoro et al. (2017)

If the number of electrons is a fraction of the number of connected bands (connectivity) forming an EBR, then the system is a symmetry-enforced semimetal. The EBR method allows us to easily identify candidate semimetallic materials. As an amusing fact, we find that the largest possible number of connected bands in an EBR is 24 and hence the smallest possible fraction of filled bands in a semimetal is 1/24. If, however, our graph analysis reveals an instance where the number of connected bands is smaller than the total number of bands in the EBR, we conclude that a $\mathbf{k}$-space description exists but a Wannier one does notSoluyanov and Vanderbilt (2011, 2012); Read (2017), 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 graph enumeration. We thus reformulate the momentum-space approach to topological indices as a real-space obstruction to 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 topological crystalline insulators. This leads to previously unrecognized large classes of TIs. We show the power of our approach by 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 piece together groups of bands from distinct points in the BZ. Consider a $D$-dimensional crystalline material invariant under a SG $G$, containing elements of the form $\{R|\mathbf{d}\}$, where $R$ is a rotation or rotoinversion, and $\mathbf{d}$ is a translation. A Bravais lattice of translations is generated by a set of linearly independent translations $\{E|\mathbf{t}_{i}\}$, $i=1\ldots D$, where $E$ represents the identity. Each $\mathbf{k}$ vector in the reciprocal lattice is left invariant by its little group, $G_{\mathbf{k}}\subset G$, and the Bloch wavefunctions $|u_{n}(\mathbf{k})\rangle$ transform under a sum of irreps of $G_{\mathbf{k}}$; bands at high-symmetry $\mathbf{k}$-vectors will have (non-accidental) degeneracies equal to the dimension of these representations (reps). For spinless or spin-orbit free/coupled systems, these are ordinary linear/double-valued reps. Away from high symmetry points, degeneracies are reduced and bands disperse according to conventional $\mathbf{k}\cdot\mathbf{p}$ theory.

Consider two different high-symmetry $\mathbf{k}$ vectors, $\mathbf{k}_{1}$ and $\mathbf{k}_{2}$, and a line $\mathbf{k}_{t}=\mathbf{k}_{1}+t(\mathbf{k}_{2}-\mathbf{k}_{1})$, $t\in[0,1]$ connecting them. To determine how the irreps at $\mathbf{k}_{1}$ connect to the irreps at $\mathbf{k}_{2}$ to form bands along this line, first note that the little group, $G_{\mathbf{k}_{t}}$, at any $\mathbf{k}_{t}$ on the line is a subgroup of both $G_{\mathbf{k}_{1}}$ and $G_{\mathbf{k}_{2}}$. Thus, an irrep, $\rho$, of $\mathbf{G}_{\mathbf{k}_{1}}$ at $\mathbf{k}_{1}$ will split (subduce, or restrict) along the line to a (direct) sum of irreps $\bigoplus_{i}\tau_{i}$ of $G_{\mathbf{k}_{t}}$; symbolically (here and throughout we use “$\approx$” to denote equivalence of representations)

[TABLE]

These restrictions are referred to as compatibility relations. Heuristically, they are found by taking the representation $\rho$ and “forgetting” about the symmetry elements of $G_{\mathbf{k}_{1}}$ which are not in $G_{\mathbf{k}_{t}}$. As energy bands in a crystal do not discontinuously end, the representations, $\sigma$, of $G_{\mathbf{k}_{2}}$ at $\mathbf{k}_{2}$ must also satisfy

[TABLE]

Each representation, $\tau_{i}$, corresponds to a (group of) band(s) along the line $\mathbf{k}_{t}$; bands coming from $\mathbf{k}_{1}$ in each irrep $\tau_{i}$ must join with a group of bands transforming in the same irrep coming from $\mathbf{k}_{2}$. We refer to each set of such pairings as a solution to the compatibility relations.

Compatibility relations apply to each and every connection line/plane between each pair of $\mathbf{k}$ points in the Brillouin zone, leading to strong but factorially redundant restrictions on how bands may connect in a crystal. To construct the nonredundant solutions to the compatibility relations, we map the question to a problem in graph theory. Each irrep at the different symmetry distinct $\mathbf{k}$ vectors labels a node in a graph. In our previous example, the nodes would be labelled by $\rho,\sigma,\{\tau_{1},\tau_{2},\dots\}$ for the $\mathbf{k}_{1}$, $\mathbf{k}_{2}$, and $\mathbf{k}_{t}$ high symmetry points and line, respectively. We draw the edges of the graph by the following rules: 1. Irreps at the same $\mathbf{k}$ vector can never be connected by edges – our graph is multi-partite. 2. Nodes corresponding to irreps at $\mathbf{k}_{a}$ and $\mathbf{k}_{b}$ can be connected only if $G_{\mathbf{k}_{a}}\subseteq G_{\mathbf{k}_{b}}$ or $G_{\mathbf{k}_{b}}\subseteq G_{\mathbf{k}_{a}}$ (i.e. $\mathbf{k}$-vectors are compatible). 3. Edges must be consistent with the compatibility relations. For instance, Eq. (1) corresponds to an edge from the node labelled by $\rho$ to each node labelled by $\tau_{i}$. 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– a gargantuan task. The factorial complexity is handled by several subroutines, which ensure that the minimal set of paths in momentum space is considered. Additional filters remove redundant or isomorphic solutions to the compatibility relations. The tools of graph theory then allow us to partition the nodes of the graph (the little-group irreps) into distinct connected components (subgraphs). Each component corresponds to a connected, isolated group of bands that can describe a set of valence bands in some insulating system or protected semimetal, depending on the 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 topologically nontrival 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 topological invariants (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 of topological 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 Fourier transforming linear combinations of the Bloch wavefunctions. In an atomic limit, the Wannier functions are exponentially localized, respect the symmetries of the crystal (and possibly TR) and coincide in most cases (however, see Section V) with the exponentially localized atomic orbitals at infinite (atomic limit) separation. Under the action of the crystal symmetries, different atomic sites are distributed into orbits, belonging to Wyckoff Positions (WPs); we denote the points in a Wyckoff orbit in a unit cell as $\{\mathbf{q}_{i}\}$. In analogy with the symmetry group of a $\mathbf{k}$-vector, to each site $\mathbf{q}_{i}$ there is a finite subgroup, $G_{\mathbf{q}_{i}}$, of the full SG, $G$, which leaves $\mathbf{q}_{i}$ invariant, called the site-symmetry group. For example, the A, B sites in graphene belong to WP $2b$ (the multiplicity $2$ refers to the number of symmetry-related sites in the unit cell); its site-symmetry group is isomorphic to $C_{3v}$. Wannier functions at each site $\mathbf{q}_{i}$ transform under some rep, $\rho_{i}$, of $G_{\mathbf{q}_{i}}$. Crucially, through the mathematical procedure of induction, the real-space transformation properties of these localized Wannier functions determine the little group reps of the bands at every point in the BZ: the action of 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 $\mathbf{k}$ dependent matrix rep of all symmetry elements. This restricts to reps of the little group of each $\mathbf{k}$-vector. Following Zak Zak (1982), we refer to this as a band representation (BR), $\rho_{iG}$, inducedFulton and Harris (2004) in the space-group $G$ by the rep $\rho_{i}$ of $G_{\mathbf{q}_{i}}$:

[TABLE]

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 $\rho_{G}$ is elementary if and only if it can be induced from an irreducible representation $\rho$ of a maximal (as a subgroup of the space group. See the SM) site-symmetry group $G_{\mathbf{q}}$ which does not appear in a list of exceptionsBacry (1993); Michel and Zak (2001) given in the SM.

We prove a similar statement for physically elementary band representations (PEBRs) and work out their extensive list of exceptions.Elcoro et al. (2017) In analogy with irreps, an (P)EBR cannot be written as a direct sum of other (P)BRs. A BR which is not elementary is a “composite” BR (CBR). Crucially, any topologically trivial group of bands is equivalent to some (P)BR, a conclusion of Definition 1. Conversely, any topologically nontrivial group of bands cannot be equivalent 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 into an atomic limit, exponentially localized, and transform under a BR. In a topological material, the Wannier functions for the valence (group of) bands either 1. fail to be exponentially localizable, or 2. break the crystal symmetry. An example of 1 is the Chern insulator: a nonvanishing Chern number is an obstruction to exponentially localized Wannier functions.Brouder et al. (2007) The Kane-Mele model of grapheneKane and Mele (2005) (Section IV), is an example of 2: in the ${Z}_{2}$ nontrivial phase, exponentially localized Wannier functions for the valence bands necessarily break TR symmetrySoluyanov and Vanderbilt (2011) (when the valence and conduction bands are taken together, atomic-like Wannier functions can be formed). Hence, in order to transition 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-symmetry group) for all $230$ SGs, with and without SOC and/or TR symmetry. This data allows us to enumerate all topologically trivial band structures. In an accompanying publicationElcoro et al. (2017) we describe the myriad group-theoretic data (subduction tables, etc) for each of the $5646$ EBRs and $4757$ PEBRs that we find. To generate this data, we generalized the well-known induction algorithm based on Frobenius reciprocity and presented in Refs. Evarestov and Smirnov, 1997 and Aroyo et al., 2006 to the case of double-valued representations. We present the full details of the computational methods in Ref. Elcoro et al., 2017. Additionally, the data can be accessed through programs hosted on the Bilbao Crystallographic ServerBilbao Crystallogr. Server (2017). We also give a summary table of all EBRs and PEBRs in Sec. VII of the Supplementary Material

This allows us to give, for the first time, a classification of TIs that is both descriptive and predictive. Rather than providing a classification (with a topological index, for instance) of the topological phases in each space group, divorced from predictive power, we instead formulate a procedure to determine whether a band structure is topologically trivial or topologically nontrivial, and enumerate the possible non-trivial band structures for each space group. Given the band structure of a material, we can compare isolated energy bands to our tabulated list of (P)EBRs. Any group of bands that transforms as a (P)EBR is topologically trivial. Those groups of bands that remain are guaranteed to be topologically nontrivial. Conversely, knowing just the valence orbitals and crystal structure of a material, we can immediately determine under which – if any – EBRs the bands near the Fermi level transform. If graph theory reveals that these EBRs can be disconnected, we deduce that this phase is topological. While a disconnected EBR itself serves as a topological index, standard techniques can be used to diagnose which (if any) of the more standard K-theoretic topological indicesFreed and Moore (2013) are nontrivial. In the subsequent sections, we outline this recipe, and present hundreds of new predicted topological materials.

Our identification of topological crystalline phases also goes beyond recently proposed classifications based on symmetry eigenvalues of occupied bands in momentum space, first proposed in Ref. Kruthoff et al., 2016, and applied in a modified form to three dimensional systems in Ref. Po et al., 2017. A shortcoming common to both methods is that while they produce a list of topological indices for each space group, they provide no insight into how to find or engineer materials in any nontrivial class. Second, in constrast to the claims of Ref. Po et al., 2017, by focusing only on eigenvalues in momentum space rather than the real-space structure of Wannier functions (or equivalently, the analytic structure of Bloch functions in momentum space), essential information about the topological properties of certain band structures is lost. For instance, the topological phases of SG $P6mm$ ($183$) presented in the Supplementary Material fall outside the scope of Ref. Po et al., 2017. Viewed in this light, our classification based on band representations generalizes the notion of a topological eigenvalue 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, distinguished by the number of relevant EBRs at the Fermi level when transitioning from a topologically trivial to a nontrivial phase. If, in a trivial phase, the Fermi level sits in a single (P)EBR, the material is necessarily a semimetal. If tuning an external control parameter (strain, SOC, etc), opens a gap at the Fermi level, the material necessarily becomes topological. This is because if an (P)EBR splits into a disconnected valence and conduction band, then neither the valence nor the conduction band can form BRs: only both together form a (P)EBR. This situation occurs exactly when an EBR can be consistently realized in a disconnected way in the BZ. This is precisely the sort of task suited for the graph-theoretic machinery of Section II! The archetypal example for this behavior is the quantum spin Hall transition in the Kane-Mele model of graphene with both next nearest neighbor “Haldane”Haldane (1988) and Rashba SOC, which we illustrate schematically in Fig. 1. This model hosts two spinful $p$-orbitals per hexagonal unit cell, for a total of four bands forming an EBR. In the Rashba SOC regime, all four bands are connected and the material is a gapless semimetal. Turning up Haldane SOC opens a band gap. By the preceding analysis, this gap must be topological. In Ref. Elcoro et al., 2017, we give all (hundreds of) cases – each specified by an orbital, Wyckoff position, and SG – where this can occur. We give the full details in the SM

The second class of TI’s is defined by the presence of more than one relevant EBR at the Fermi level. The trivial phase 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 involving more than two EBRs can be resolved into a sequence of pairwise transitions. A topological phase transition occurs when a gap closes and reopens after a band inversion, such that neither the valence bands nor the conduction bands form a BR. In the trivial phase, the little group irreps of the filled bands at each $\mathbf{k}$ point are those of the valence band EBR. After the topological phase transition, the little group irreps at each $\mathbf{k}$ point are not consistent with an EBR. This mechanism describes the zeitgeist $3D$ TI Bi2Se3.Zhang et al. (2009); Xia et al. (2009) Without, or with very small, SOC, Bi2Se3 is a trivial insulator; its valence and conduction bands transform in two distinct EBRs. Increasing SOC pushes the bands together; at a critical strength, the gap between the valence and conduction bands closes at the $\Gamma$ point of the BZ. Above this critical value, a gap reopens, with certain states (labelled by irreps of $G_{\Gamma}$) exchanged between the valence and conduction bands. Ultimately, neither the valence bands nor the conduction bands transform as EBRs and the insulator is topological as per Def. 1Winkler et al. (2016). If, on the other hand, a full gap does not reopen after band inversion, then we are left with symmetry-protected semimetal a la Cd3As2Liu et al. (2014a) or Na3BiLiu et al. (2014b).

When the phase transition is driven by SOC, we label the classes by $(n,m)$, where $n$ is the number of EBRs at the Fermi level in the trivial phase (without SOC) and $m$ is the number in the topological phases (after SOC is turned on). These phases are indicated in Table 1, 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 constructed all 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 localized symmetric Wannier functions (topologically trivial insulators.) It follows that any other group of valence bands in an insulator necessarily constistute a TI. Importantly, our theory also shows that there exist different classes of topologically trivial insulators, which cannot be adiabatically continued to one another. We now link this important fact 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 orbitals from different atomic sites can hybridize to form bonding and antibonding “molecular” orbitals, centered away from any individual atomHoffmann (1987). In a crystal formed of these tightly bound molecular (rather than atomic) units, the valence and conduction BRs are induced from the (generically maximal) WPs of the molecular orbitals, rather than from the atomic orbitals; consequently, the valence and conduction band Wannier functions are localized at the molecular orbital WPs, away from the atoms.

Thus, orbital hybridization, when viewed in the solid-state, represents the required transition between two symmetry distinct atomic limit phases. In both phases localized, symmetric Wannier functions exist; the distinction lies in where the orbitals sit in the atomic limitZak (1989); King-Smith and Vanderbilt (1993). 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 topologicalRyu and Hatsugai (2002); Bernevig and Hughes (2013), but we refer to it as an “obstructed” atomic limit, since it is also described by localized Wannier statesKivelson (1982); Read (2017). The prototypical example is the $1D$ Su-Schrieffer-Heeger (Rice-Mele) chain, whose two phases are distinguished by their hybridization pattern, an electric dipole moment that is [math] ($\frac{1}{2}$) in the trivial (nontrivial) phase. Pumping between two different atomic limits always leads to a nontrivial cycle with observable transport quantities.Rice and Mele (1982); Atala et al. (2013); Nakajima et al. (2016) A similar phenomenon is observed in the newly discovered quadrupole insulators.Benalcazar et al. (2016) Signatures of the obstructed atomic limit also appear in the real-space entanglement structure of the insulating ground state in finite systemsFang et al. (2013), and these signatures persist to systems containing only a single molecule.Tubman and Yang (2014)

We now describe orbital hybridization with EBRs. The valence and conduction bands are each described by an EBR. Taken together, the two EBR’s comprise a CBR. In both the first and the second atomic limit, as well as at the critical point between them, the CBR does not change, although the individual EBR’s of the valence and conduction bands do. We denote the CBR in the first atomic limit as $\sigma_{v}\uparrow G\oplus\sigma_{c}\uparrow G$, where $\sigma_{v}$ and $\sigma_{c}$ are irreps of the site symmetry group, $G_{a}$, of the atomic sites. In the second atomic limit, the CBR is $\rho_{v}\uparrow G\oplus\rho_{c}\uparrow G$, where $\rho_{v}$ and $\rho_{c}$ are irreps of the site symmetry group, $G_{m}$, of the molecular sites. As we show in the SM, a symmetry-preserving transition can only happen when the two site symmetry groups, $G_{a}$ and $G_{m}$, have a common subgroup $G_{0}$ with a representation $\eta$ such that

[TABLE]

This equation indicates that there is a line joining the atomic sites, and that Wannier functions localized along the line give the same set of bands as those localized at either endpoint. Because of this, the Wannier functions for the valence band can move from the atomic to the molecular sites while preserving all symmetries upon passing through a critical point. In the SM, we illustrate this using the example of $sp$ orbital hybridization. Note, furthermore, that we can define an analogous notion of an obstructed atomic limit for systems lacking translational symmetry, using properties of the point group only. In this way, the preceding discussion generalizes straightforwardly to finite-sized crystals, molecules, and even quasicrystals. In the latter case, obstructions to the naive atomic limit are precisely covalent 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 Inorganic Crystal Structure Database (ICSD)Fachinformationszentrum Karlsruhe (2016) – 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 distinguish between the number of EBRs at the Fermi level, i.e., the two classes defined in Section IV and summarized in Table 1. We further distinguish between cases with and without SOC, which can be treated separately by existing ab-initio methods. The SOC strength can be viewed as a control parameter driving the topological phase transition.For EBRs without SOC, we count bands on a per-spin basis: all states are doubly degenerate. Further new topological semimetals, such as a series of ones at filling -1/8, can be obtained by our method.

VI.1 Single PEBRs at the Fermi Level

As described in Section IV and Table 1, a $(1,1)$ type TI occurs when a single PEBR is realized as a sum of two band groups disconnected in momentum space. Here, we utilize the fact that $(1,1)$ TIs can be realized by band representations induced from $1D$ site symmetry group irreps that are EBRs but not PEBRs. In this case, the Wannier 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 is necessarily a TI.

An example is furnished by lead suboxide Pb2OFerrari (1926), a material whose topological properties were until now unexplored. As discussed in the Supplementary Material, this is a non-symmorphic cubic crystal in the space group $Pn\bar{3}m$ ($224)$. Although metallic, this material features a topologically disconnected PEBR far below the Fermi level, shown in Fig. 2a. However, we can consider the application of $z-$axis uniaxial strain, under which the crystal symmetry is lowered to the tetragonal subgroup $P4_{2}/nnm$ (134) of the original space group. There are fewer symmetry constraints imposed on the band structure in this space group, and in particular degeneracies protected by threefold rotational symmetry will be broken. This allows for a gap to open at the Fermi level, leading us to predict that strained Pb2O will be a topological insulator with a small Fermi pocket, as shown in Fig. 2b. As an aside, we note that this analysis shows that by using group-subgroup relations, we can make predictions about the topological character of strained systems when the unstrained band structure is known.

Cu3SbS4, a candidate TIPfitzner and Reiser (2002); Wang et al. (2011), is also an example of a $(1,1)$ type material. In Fig. 2c we show the calculated band structure. The states near the Fermi level form a single EBR coming from the Cu $d$-orbital electrons. With SOC, the EBR is gapped, leading to topologically nontrivial valence and conduction bands. A trivial spurious EBR also lies energetically within the topological gap (shown in black in the inset to Fig. 2c), causing the effective transport gap to be smaller than the topological gap (similar to HgTe). There are $35$ additional materials in this Cu2ABX4 class of materials.

We use these considerations to design a systematic method to search for $(1,1)$ type TI’s. First, we identify all disconnected PEBRs from our data paper Ref. Vergniory et al., 2017, which is organized by WP and site-symmetry irrep; in the SM we indicate which orbital types give rise to these representations. Next, we cross-reference with the ICSDFachinformationszentrum Karlsruhe (2016), which yields a list of candidate materials. This list can be further reduced by restricting to semi-metals (since an insulator would 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 near the Fermi level. Carrying out this procedure led us to identify the Cu2ABX4 material class introduced in the previous paragraph. We also provide a listsup of materials guaranteed to be semi-metals by the minimal-insulating-filling criteriaWatanabe et al. (2016) (a sufficient, but far from necessary condition for a semi-metal).

VI.2 Multiple PEBRs

As described in Section IV, a CBR can be disconnected in such a way that neither the valence nor the conduction bands form PEBRs. When these are double-valued (spinful) BRs, we classify them by whether the BR is composite or elementary without SOC. In the first case, a single, connected, SOC-free EBR decomposes after turning on SOC into 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*-1* square nets and related structures, which we discuss further in the SM. The relevant states at the Fermi level are the Bi $p_{x}$ and $p_{y}$ orbitals, which induce a single EBR when SOC is neglected. These materials are filling-enforced semimetals with a symmetry-protected nodal (degenerate) line at the Fermi 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 SrZnSb2 and ZrSnTeXu et al. (2015); Lou et al. (2016).

A similar materials search program to that described in the previous subsection can be implemented for the $(1,2)$ type of Table 1. SOC-free EBRs which decompose into CBRs with SOC can be quickly identified from Ref. Elcoro et al., 2017. With these in hand, a list of materials with the proper orbitals and occupied WPs can be compiled from the ICSDFachinformationszentrum Karlsruhe (2016). Finally, an SOC-free electron counting analysis will reveal which of these materials are semi-metallic without SOC. These are prime candidates to become TIs when SOC is included. This systematic search has led us to identify over 300 $(1,2)$ materials in SG $P4/nmm$ ($129$)Xu et al. (2015); Lou et al. (2016); Schoop et al. , as well as 58 new candidate TIs in SG Pnma ($62$). SG Pnma $(62)$ arises from an in-plane distortion of SG $P4/nmm$ ($129$), and again demonstrates that our group-theoretic approach allows us to predict the topological character of materials upon structural distortion. We present these materials in the SM.

Finally, we discuss the known TIs Bi2Se3 and KHgSb. In these materials, without SOC the band representation at the Fermi level is gapped and composite. With infinitesimal SOC, the band representation remains gapped and composite. A topological phase transition occurs only when SOC is strong enough to drive a band inversion that exchanges states with distinct little group reps at $\Gamma$ between the conduction and valence bands. Because such a transition depends on the strength of SOC – unlike in the $(1,1)$ and $(1,2)$ cases, where the transition is at infinitesimal SOC – Bi2Se3 and KHgSb are described by our method but could not be unequivocally predicted.

VI.3 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 and semimetals. Going beyond standard methods of counting multiplicities of occupied WPsHoffmann (1987), we can predict symmetry-protected semimetals by looking for partially-filled, connected EBRs induced from high-dimensional site-symmetry representations. In this way we have found the A15B4 familyMorris et al. (2014) of sixteen-fold connected metals in SG $I\bar{4}3d$ (220) with A$=$Cu,Li,Na and B$=$Si,Ge,Sn,Pb. Through charge-transfer, the sixteen bands in this PEBR are $7/8$ filled; this exotic filling holds promise for realizing novel physics when interactions are included. The band structure for Li15Ge4 is shown in Fig. 2d. There is a symmetry-protected threefold degeneracy near the Fermi level at the $P$ pointBradlyn et al. (2016) Finally, we calculate that Cu3TeO6 in SG $Ia\bar{3}$ (206)Falck et al. (1978) has a half filled, connected twenty-four band PEBR at the Fermi level when interactions are neglected. Our EBR theory reveals that this is the highest symmetry-enforced band connectivity in any SG.

VI.4 Systematic materials search: Summary

While the materials presented above represent the proof-of-principle for our material search strategy, a full, systematic search of the entire materials database based on our criteria reveals a myriad of new topological insulator and 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, and verified with ab-intio DFT calculations. In SG P$\bar{3}m1$ ($164$) we predict that CeAl2Ge2, CeISi, BiTe, and Nb3Cl8 will be topological insulators. In SG $P4/mmm$ ($123$) we have LiBiS2 and AgSbTe2. A particularly promising family of topological materials is given by TiAsTe, ZrSbTe, HfSbTe, Hf3Ni4Ge4, Sr3Li4Sb4, Ba2Bi3, and Ba3Al2Ge2 in SG $I/mmm$ ($71$). Additionally, we find NaAu2 in SG $Fd\bar{3}m$ ($227$), LaPd2O4 in SG $I4_{1}/a$ ($88$), BaGe2Ru2 in SG $Fddd$ ($70$), Ni2Ta2Te4 in SG $Pbam$ ($53$), Ag2Ca4Si6 in SG $Fmmm$ ($69$), Ta2Se8I in SG $I422$ ($97$), SnP in SG $I4mm$ ($107$), and Tl4CuTe3 in SG $I4/mcm$ ($140$) 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 Quantum Chemistry. 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 theory is descriptive and predictive: we can algorithmically search for and predict new TIs and semimetals. In a series of accompanying Data papers, we present the group-theoretic data and graph-theoretic algorithms necessary to deduce the conclusions of Sections II and III, and to implement the materials search described in Section VI. By taking the ideas 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 and Bloch wavefunctions, allows for a full understanding of noninteracting solids, which we have only begun to explore in this work. Furthermore, our emphasis on the symmetry of localized orbitals opens a promising avenue to incorporate magnetic groups or interactions into the theory of topological materials.

Data Availability: All data supporting the conclusions of this work is hosted on the Bilbao Crystallographic Server (http://cryst.ehu.es). All information about EBRs, PEBRs, and their connectivity graphs can be accessed via the BANDREP applicationBilbao Crystallogr. Server (2017). The algorithms used to generate this data, as well as a guide to the use of all relevant programs, can be found in the accompanying data papers, Refs. Elcoro et al., 2017; Vergniory et al., 2017.

Acknowledgements: BB would like to thank Ivo Souza, Richard Martin, and Ida Momennejad for fruitful discussions. MGV would like to thank Gonzalo Lopez-Garmendia for help with computational work. BB, JC, ZW, and BAB acknowledge 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 the hospitality and support of the École Normale Supérieure and Laboratoire de Physique Théorique et Hautes Energies. The work of MVG was supported by FIS2016-75862-P and FIS2013-48286-C2-1-P national projects of the Spanish MINECO. 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 and BAB, as well as part of the development of the initial theory and further ab-initio work, were supported by the Department of Energy de-sc0016239, Simons Investigator Award, the Packard Foundation, and the Schmidt Fund for Innovative Research. The development of the practical part of the theory, tables, some of the code development, and ab-initio work was funded by NSF EAGER Grant No. DMR-1643312, ONR - N00014-14-1-0330, and NSF-MRSEC DMR-1420541.

Author Contributions: BB, LE, JC, MGV, and ZW contributed equally to this work. BB, JC, ZW and BAB provided the theoretical analysis, with input from CF. JC developed specific models to test the theory. LE and MIA performed the computerized group-theoretic computations. BB, LE and MGV devised and developed the graph algorithms, 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-principles calculations.

Author Information: Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to bernevig@princeton.edu.