Topological materials discovery by large-order symmetry indicators

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Science Advances  08 Mar 2019:
Vol. 5, no. 3, eaau8725
DOI: 10.1126/sciadv.aau8725


Crystalline symmetries play an important role in the classification of band structures, and their richness leads to various topological crystalline phases. On the basis of our recently developed method for the efficient discovery of topological materials using symmetry indicators, we explore topological materials in five space groups (SGs), which are diagnosed by large-order symmetry indicators (ℤ8 and ℤ12) and support the coexistence of several kinds of gapless boundary states in a single compound. We predict many candidate materials; some representatives include Pt3Ge (SG140), graphite (SG194), XPt3 (SG221, X = Sn, Pb), Au4Ti (SG87), and Ti2Sn (SG194). As by-products, we also find that AgXF3 (SG140, X = Rb, Cs) and AgAsX (SG194, X = Sr, Ba) are good Dirac semimetals with clean Fermi surfaces. The proposed materials provide a good platform for studying the novel properties emerging from the interplay between different types of boundary states.


Since the discovery of two-dimensional (2D) and 3D topological insulators (TIs), band topology in condensed matter materials has attracted broad interest owing to their rich scientific implications and potential for technological applications (1, 2). Described by ℤ2 topological invariant(s), time-reversal (T) invariant TIs are characterized by an insulating gap in the bulk and T-protected gapless states on the boundary of the system (1, 2). Inspired by the discovery of TIs, it was realized that symmetries play a key role in the classifications of topological phases. On the basis of the absence or presence of T, particle hole, or chiral symmetry, insulators and superconductors have been classified under the so-called 10-fold way (3).

In addition to the aforementioned internal symmetries, the topological classification of band structures has also been extended to include crystalline symmetries (46). Because of the vast array of crystal symmetries [encapsulated by the 230 crystalline space groups (SGs)], a massive number of topological crystalline phases (TCPs) have been proposed, such as mirror Chern insulators (7), quantized electric multipole insulators (8), high-order TIs (9, 10), hourglass fermions (11), nodal-chain metals (12), and (semi-)metals with unconventional quasiparticles arising from threefold (or higher) band degeneracies (13).

Despite the large number of theoretically proposed TCPs, the discovered topological compounds represent a very small fraction of the experimentally synthesized materials tabulated in structure databases (14). Such apparent scarcity of topological materials originates from the theoretical difficulty in exhaustively computing topological band invariants in first-principles calculations, which becomes increasingly time consuming because of the expanding set of identified invariants (47, 9, 11, 1523). Hence, the prediction of any realistic topological materials is typically taken as a big achievement (1, 2, 6, 7, 1113, 2426).

Recent theoretical advancement has greatly reshaped the landscape of materials discovery. By exploiting the mismatch between the real- and momentum-space descriptions of band structure, novel forms of band topology in the 230 SGs for nonmagnetic compounds (27, 28) and the 1651 magnetic SGs for magnetic materials have been proposed (29). A main advantage of the formalism of symmetry-based indicators of band topology (27) is its compatibility with first-principles calculation: In stark contrast to conventional target-oriented searches, our algorithm does not presuppose any specific phase of matter. Based on symmetry representations, which can be readily computed using standard protocols, one can quickly discern topological (semi-)metals, TIs, and topological crystalline insulators from the database (30). The high efficiency of our method has been demonstrated in (30), in which we discuss many topological materials discovered on the basis of their nontrivial index in space groups with ℤ2 or ℤ4 strong factor, XBSs, in the symmetry indicator (SI) group.

In this work, we focus on SGs with XBSs=8 or ℤ12. One particularly interesting aspect of the materials candidates we present in this work is the coexistence of topological surface states originating from bulk-boundary correspondence (1, 2) dictated by various kinds of spatial symmetries (27, 31, 32). These SIs are realized in SGs with a high degree of coexisting symmetries, such as (roto-)inversion, mirror reflection, screw, and glide. There are in total 12 and 6 SGs with strong ℤ8 and ℤ12 SI factor groups, respectively (27). Focusing on five SGs with ℤ8 or ℤ12 strong SI group (SGs87,140,221,191,and194), we search for interesting TCPs in a single sweep of a structure database (14) using the method delineated in (30). We only consider spin orbit (SO)–coupled nonmagnetic materials with ≤ 30 atoms in their primitive unit cell. We find a large number of TCPs with reasonably clean Fermi surfaces. In the following, we present and discuss six representative topological crystalline insulators (TCIs) and list others in Tables 1 and 2. Four good Dirac semimetal candidates are discussed at the end.

Table 1 Topological crystalline (TC) insulating materials for SGs87,140, and 221.

These SGs all own the same strong SI factor group, ℤ8, but with different other weak SI factor groups. The numbers in the parenthesis following the name of material are the nonvanishing SI in the corresponding XBS. The SI is obtained by qiCi mod Ci, where ai has a common factor larger than 1, which corresponds to the subscript of the factor groups of XBS. The blue color denotes the materials carefully discussed in this work.

View this table:
Table 2 TC insulating materials for SGs191 and 194.

These SGs all own the same strong SI factor group, ℤ12, but with different other weak SI factor groups. The numbers in the parenthesis following the name of material are the nonvanishing SI in the corresponding XBS. The SI is obtained by qiCi mod Ci, where ai has a common factor larger than 1, which corresponds to the subscript of the factor groups of XBS. The blue color denotes the materials carefully discussed in this work.

View this table:


We begin by providing a brief review on topological materials discovery using SIs (30). In this paradigm, the topological properties of materials can be assessed by computing the representations of the filled energy bands at high-symmetry momenta, which is a standard protocol in band structure calculations. More concretely, the representation content is encoded in a collection of integers, nkα, which can be written as a formal vector: n=(ν,nk11,nk12,), where v is the total number of the filled energy bands, the subscript k1, k2, …, kN denotes the high-symmetry point (HSP) in the Brillouin zone, the superscript 1, 2, …, αi, … refers to the irreducible representation (irrep) of little group at ki point (Gki), and nkiαi means the number of times an αi irrep of Gki appears among the filled bands.

The set of vectors n forms an abelian group (27, 33). Moreover, for every SG, there exists dAI atomic insulator (AI) basis vectors (ai, i = 1, 2, …, dAI) containing information of the group structure for the SI, denoted by XBS in (27), according to the possible common factor Ci for ai (27). One can always expand any vector n with respect to the AI basis vectors ai: n=i=1dAIqiai. The expansion coefficients of n on the AI basis can be classified into three cases (30). Case 1: The expansion coefficients qis are all integers; such materials might be adiabatically connected to a trivial AI, so we do not consider materials in this case. Case 2: The expansion coefficients qi are not all integers, but all qiCi are integers; such materials are necessarily topological, and the results of (qiCi mod Ci) give the nonvanishing SI (30). Case 3: The qiCis are not all integers; such systems are (semi-)metallic. Specifically, if nkiα is noninteger, then there is band crossing happening at ki point; on the other hand, if all the nkα are integers, then there must be band crossing in the high-symmetry line or plane (30).


We now describe the promising TCP materials candidates we discovered. We first perform our materials search in the nonsymmorphic SG140 (I4/mcm), which has seven AI basis vectors: aiSG140,i=1,2,,7 (all the AI bases of the SGs in this work are explicitly given in the Supplementary Materials). Only two AI basis vectors (we label them as a6SG140 and a7SG140) have nontrivial common factors: 2 and 8, respectively. Correspondingly, the SI group of SG140 is XBSSG140=2×8. We list the relatively good materials belonging to case 2 in Table 1. In the following, we take Pt3Ge as the example to analyze the detailed topological properties.

Pt3Ge (ICSD[14] 077962) crystallizes in the body-centered tetragonal structure, where Ge occupies the 4b Wyckoff position, and Pt occupies two sets of inequivalent sites in the 4a and 8h Wyckoff positions. There are in total 68 valence electrons in the primitive unit cell. On the basis of ab initio calculation, we calculate the irrep multiplicities nkα for all the HSPs and all the corresponding irreps α for the 68 valence bands. We then expand this calculated vector n on the seven AI basis vectors n=i=17qiaiSG140 and obtain q=(8,0,0,1,2,1,12). Thus, this material belongs to case 2 and is a TCI, with SI being (0,4). As seen from the electronic band plot in Fig. 1A, this material has large direct gaps through the k path.

Fig. 1 Electronic band plots of TCIs.

(A) Electronic band plot of TCI Pt3Ge within SG140. (B) Electronic band plot of TCI graphite within SG194.

While from SI alone we can ascertain that Pt3Ge is a T(C)I, to resolve the concrete form of band topology it displays, we have to evaluate additional topological indices. First, we note that from the Fu-Kane parity criterion (15), one sees that the material cannot be a strong or a weak TI, and Pt3Ge must be a TCI. As discussed in (31) and (32), the band topology of a TCI can be understood in terms of a collection of invariants associated to each of the elements of the space group. When the invariant of an element is nontrivial, one finds protected surface states on suitable surface terminations. For instance, if a glide invariant is nontrivial, one finds the hourglass surface states on surfaces respecting the glide symmetry (11). For symmetries like inversion and screws, which cannot leave any point invariant on the surface, their nontrivial invariants manifest as hinge states at suitable surface termination.

For Pt3Ge, we find that the enriched inversion invariant κ1 mod 4 (31, 32) is also vanishing. Thus, this material has boundary states protected by symmetry operation containing n-fold axis (n > 1), mirror, and/or glide symmetries (31, 32). Because of the rich point symmetry operations in SG140 (whose point group is D4h), several topological phases may occur (31, 32). We thus evaluate the mirror Chern numbers for the (001) (Miller indices with respect to the conventional lattice basis vectors) and (110) mirror planes by first-principles calculations. Our ab initio results show that they are also all vanishing. As shown in (32), with the above SI and mirror Chern numbers, the glide, screw, and S4 invariants are thus nonvanishing (31, 32): It would have glide-protected hourglass surface states in (100) glide symmetric planes as the corresponding invariant is 1. The C4z-screw invariant is 1; thus, it would protect gapless hinge states along the c direction. We construct a tight binding (TB) model and fit its electronic structure, the SI, and all the topological invariants with the corresponding first-principles results. By the TB model, we demonstrate the surface hourglass band crossings as shown in the Supplementary Materials.


We also searched the materials with SG194 (P63/mmc, whose point group is D6h) in the database (14). We find that there are 52 and 254 materials belonging to cases 2 and 3, respectively. It is worth emphasizing that our results indicate that graphite (ICSD[14] 193439) is potentially a nontrivial insulator.

It is well known that graphene (i.e., monolayer of graphite) exhibits 2D massless Dirac excitation near K/K′ points (34). The SO coupling (although small) opens a topological gap [~0.0008 meV (35)], making it, in principle, a 2D TI (36). The discovery of crystalline-symmetry–protected band topology in graphite, namely, the ABABABAB… Bernal stacking of graphene, demonstrates the possibilities of discovering various topological materials even among the simplest elemental materials. We thus present a detailed discussion in the following for graphite.

The SG194 owns nine AI basis vectors aiSG194,i=1,2,,9, where only the last one has a common factor, which is 12. Thus, XBSSG194=12. The 16 valence bands in graphite are found to have the expansion coefficients q=(2,0,1,1,1,1,1,3,13) on the AI basis. Thus, the SI for graphite is 4 ∈ ℤ12. The band structure is shown in Fig. 1B, where the SO coupling opens a small gap (around 0.025 meV) at the K point according to the first-principles calculation. The Fu-Kane strong and weak topological invariants (15) are found to be all vanishing as well as that κ1 mod 4 is zero. We then calculate the (001) mirror Chern number at kz = 0 by first-principles method and find that it is −2. Thus, there would be gapless Dirac surface states in the (001) mirror symmetric planes in the line kz = 0. For another mirror symmetric plane, kz=πc, the mirror Chern number must be vanishing: for SG194, TJ (J is the inversion) preserves the sign of the mirror eigenvalue, while it reverses the sign of Berry curvature: The Berry curvature for +i mirror eigenvalue Ωz+ should satisfy TJΩz+=Ωz+=0. We construct a TB model that reproduces the ab initio band structures very well and also gives the same SI and topological invariants to demonstrate the surface states protected by (001) mirror plane in the Supplementary Materials.

To ascertain graphite’s nontrivial topology, we then calculate the (1¯20) plane’s mirror Chern number and find that it is vanishing. Then, graphite would have sixfold screw-protected hinge states (31, 32). It can also have glide- and rotation-protected surface states as dictated by the nonvanishing C211¯0 (where the superscript of the point operation part denotes the rotation axis and the subscript denotes the rotation angle) rotation invariant and (010) glide invariant (31, 32). While graphite is generally associated with small Fermi pockets, García et al. (37) proposed, based on the observation of a semiconducting gap in small samples of Bernal graphite, that these may arise from extrinsic effects. Thus, further experimental work would be of great interest.


Weak TI coexisting with TCI in PbPt3 (SG221) and Au4Ti (SG87)

The above two TC materials both have vanishing inversion and weak topological invariants. We also discover two materials, i.e., PbPt3 (ICSD[14] 648399) in SG221 and Au4Ti (ICSD[14] 109132) in SG87, which have three weak topological indices (15) νi = 1 for i = 1, 2, 3 (ν0 is vanishing for both cases); however, they have different inversion topological invariants, i.e., κ1 mod 4 (31, 32) is equal to 0 or 2, respectively.

PbPt3 crystallizes in the cubic structure with a primitive Bravais lattice. The electronic band structure is shown in Fig. 2A. The material has 34 valence electrons in the unit cell. The calculated nkα for these 34 bands can be expanded on the 14 AI basis vectors of SG221, and the expansion coefficients are q=(0,0,0,0,0,1,1,1,0,1,1,1,14,12). The last two AI basis vectors own a common factor 4 and 8, respectively. Thus, the SI is (3, 4) ∈ ℤ4 × ℤ8. On the other hand, the parity calculations show that it is a weak TI (15). To further pin down the precise topological character of the system, we also calculate the two mirror Chern numbers for (001) mirror plane (kz = 0 or πc) (31, 32) and find that they are both equal to −1. This implies that the screw invariant of 21011, as discussed in (32), is 1. Note that, as SG221 is symmorphic, the screw 21011 discussed above is not essential, in the sense that it is really a combination of the rotation C2011 combined with a lattice translation with a nonzero parallel component along the 011 rotation axis. Nonetheless, on the appropriate surface termination, which, as a whole, respects this screw symmetry, one could find protected hinge states.

Fig. 2 Electronic band plots of TCIs.

(A) Electronic band plot of PbPt3 within SG221. (B) Electronic band plot of Au4Ti within SG87.

Au4Ti crystallizes in SG87 (I4/m), where Au and Ti occupy 8h and 2a Wyckoff positions, respectively. This material is found to belong to case 2. The electronic band structure is shown in Fig. 2B. We calculate the parities and find that its strong topological invariant (15) and inversion invariant κ1 mod 4 (31, 32) are both vanishing, while ν1 = ν2 = ν3 = 1, so it is a weak TI. Besides, the newly introduced invariant Δ (31) is found to be 4 (mod 8). Our first-principles calculations also show that the mirror Chern number for the (001) plane is vanishing. Thus, it would allow glide-protected hourglass surface states in glide {M001|12120} symmetric plane (31, 32). It can also host hinge states along the (001) direction, which are protected by the (nonessential) screws {C2001|0012} or {C4001|0012} (32).

TCI Ti2Sn in SG194

Ti2Sn (ICSD[14] 182428) within SG194 is found to be a TCI. The electronic band structure plot is shown in Fig. 3. It has direct gaps everywhere, except in a small area where there are electron and hole pockets. Our calculations show that the SI is (6). Parity calculations show that the inversion invariant κ1 mod 4 (31, 32) is 2, while the strong and weak topological invariants (15) ν0,1,2,3 are all vanishing. From first-principles calculation, we find that the mirror Chern number for the (1¯20) plane is −4. This high mirror Chern number indicates that there should be multiple Dirac cones in the (1¯20) mirror symmetric plane. To identify the band topology, we also calculate the mirror Chern number of the (001) mirror plane, which is found to be vanishing. Thus, it can accommodate hourglass surface states in {M010|0012} or {M010|12012} glide symmetric planes (31, 32). C2 around (010) can also protect surface Dirac cones (31, 32). Besides, inversion and screw {C6001|0012} can protect hinge states in corresponding hinges, satisfying the corresponding symmetries (31, 32).

Fig. 3 Electronic band plot of TCI Ti2Sn within SG194.


Other than the TCIs, our method can also filter out topological (semi-)metals as by-products when the expansion coefficients belong to case 3. By further requiring relatively clean Fermi surfaces, we identify AgXF3 (X = Rb, Cs; SG140; ICSDs[14] 023153,023154) as good Dirac semimetals, with Dirac points pinned down to two HSPs (P and N) and AgAsX (X = Sr, Ba; SG194; ICSDs[14] 049742,008278) as Dirac semimetals with symmetry-protected band crossing at the high-symmetry line, as shown in Fig. 4. These two materials families realize the two subcases within case 3 that we discussed. For the AgXF3 family, the HSPs P and N both have only one 4D irrep, while the filling cannot be divided by 4. The filling-enforced Dirac points at P or N are subjected to more symmetry restrictions than those for the Dirac points in the high-symmetry line, and consequently, the Dirac dispersion is more isotropic. For the AgAsX family, in the high-symmetry line Γ-A, the Δ7 and Δ9 bands cross each other, resulting in a Dirac point protected by C6v. It is worth pointing out that for AgAsX, the Fermi level exactly threads the Dirac point.

Fig. 4 Electronic band plots of Dirac semimetals.

(A) AgCsF3 within SG140 owns Dirac points pinned down at P and N points. (B) AgAsBa within SG194 has a Dirac point lying in the high-symmetry line ΓA. The Dirac point is protected by C6v, and the band crossing arises from two twofold degenerate bands with different irreps (Δ7 and Δ9).


In this work, on the basis of our newly developed algorithm (30), we search for topological materials indicated by ℤ8 and ℤ12 strong factors in the SI groups. Focusing on SG s 87, 140, 221, 191, and 194, we predict many materials, which exhibit coexistence of various gapless boundary states due to the rich combination of various symmetry operators in these highly symmetric SGs. Breaking the symmetry operation directly affects (move or even gap) the gapless topological boundary state; thus, one may easily tune the novel properties of these predicted topological materials through strain or boundary decoration.

It is worth mentioning that the electronic topological phenomenon is widespread in real materials, and as shown in fig. S3, most of the materials in the five SGs we scanned belong to topological phases indicated by cases 2 and 3. Here, we only discuss the materials with clean Fermi surfaces, since in these materials we expect the transport properties to be dominated by the topologically nontrivial states. Our scheme also finds some good metals with big Fermi surfaces having nonzero SI. One good example is MgB2. It is interesting to contemplate on the possible interplay between its superconductivity (38) and band topology.

We hope that our proposed materials will enrich the set of realistic topological crystalline materials and stimulate related experiments. With the demonstrated efficiency, our method (30) can be used for a large-scale systematic search of the entire materials database, which could lead to the discovery for many more new topological materials.


The electronic band structure calculations were carried out using the full potential linearized augmented plane-wave method as implemented in the WIEN2k package (39). The generalized gradient approximation with Perdew-Burke-Ernzerhof (40) realization was adopted for the exchange-correlation functional.


Supplementary material for this article is available at

Section S1. First-principles calculated parities

Section S2. S4 invariant materials

Section S3. Details of calculating mirror Chern numbers by first-principles method

Section S4. Details of the TB model and the glide/mirror-protected surface states

Section S5. AI basis vectors

Section S6. Materials statistics

Table S1. Ab initio calculated parities.

Table S2. Ab initio calculated κ4 for body-centered lattice.

Table S3. Ab initio calculated κ4 for primitive lattice.

Table S4. AI basis vectors in this work for SGs 87, 140, and 221.

Table S5. AI basis vectors in this work for SGs 191 and 194.

Fig. S1. TB fitting of Pt3Ge and its surface states.

Fig. S2. TB fitting of graphite and its surface states.

Fig. S3. Materials statistics.

Source code

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.


Acknowledgments: Funding: F.T. and X.W. were supported by the National Key R&D Program of China (nos. 2018YFA0305704 and 2017YFA0303203), the NSFC (nos. 11525417, 11834006, 51721001, and 11790311), and the excellent programme in Nanjing University. F.T. was also supported by the program B for Outstanding PhD candidate of Nanjing University. X.W. was partially supported by a QuantEmX award funded by the Gordon and Betty Moore Foundation’s EPIQS Initiative through ICAM-I2CAM, grant GBMF5305, and the Institute of Complex Adaptive Matter (ICAM). A.V. was supported by NSF DMR-1411343, a Simons Investigator grant, and the ARO MURI on TIs (grant W911NF- 12-1-0961). H.C.P. was supported by a Pappalardo Fellowship at MIT. Author contributions: X.W., A.V., and H.C.P. conceived and designed the project. F.T. performed ab initio calculations. All authors contributed to the writing and editing of the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Our codes of symmetry-indicator method are present in Additional data related to this paper may be requested from the authors.

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