## Abstract

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 (_{8} and ℤ_{12}) and support the coexistence of several kinds of gapless boundary states in a single compound. We predict many candidate materials; some representatives include Pt_{3}Ge (*X*Pt_{3} (*X* = Sn, Pb), Au_{4}Ti (_{2}Sn (*X*F_{3} (*X* = Rb, Cs) and AgAs*X* (*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.

## INTRODUCTION

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 ℤ_{2} 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 (*4*–*6*). Because of the vast array of crystal symmetries [encapsulated by the 230 crystalline space groups (*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 (*4*–*7*, *9*, *11*, *15*–*23*). Hence, the prediction of any realistic topological materials is typically taken as a big achievement (*1*, *2*, *6*, *7*, *11*–*13*, *24*–*26*).

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 *27*, *28*) and the 1651 magnetic *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 ℤ_{2} or ℤ_{4} strong factor,

In this work, we focus on _{12}. 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 _{8} and ℤ_{12} SI factor groups, respectively (*27*). Focusing on five _{8} or ℤ_{12} strong SI group (*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.

## REVIEW OF SYMMETRY INDICATORS

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, *v* is the total number of the filled energy bands, the subscript **k**_{1}, **k _{2}**, …,

**k**

_{N}denotes the high-symmetry point (HSP) in the Brillouin zone, the superscript 1, 2, …, α

_{i}, … refers to the irreducible representation (irrep) of little group at

**k**

_{i}point (

_{i}irrep of

The set of vectors **n** forms an abelian group (*27*, *33*). Moreover, for every *d*_{AI} atomic insulator (AI) basis vectors (**a**_{i}, *i* = 1, 2, …, *d*_{AI}) containing information of the group structure for the SI, denoted by *X*_{BS} in (*27*), according to the possible common factor *C*_{i} for **a**_{i} (*27*). One can always expand any vector **n** with respect to the AI basis vectors **a**_{i}: **n** on the AI basis can be classified into three cases (*30*). Case 1: The expansion coefficients *q*_{i}*C*_{i} mod *C*_{i}) give the nonvanishing SI (*30*). Case 3: The **k**_{i} point; on the other hand, if all the *30*).

## ℤ_{8} NONTRIVIAL TCI: Pt_{3}Ge

We now describe the promising TCP materials candidates we discovered. We first perform our materials search in the nonsymmorphic *I*4/*mcm*), which has seven AI basis vectors: _{3}Ge as the example to analyze the detailed topological properties.

Pt_{3}Ge (ICSD[14] 077962) crystallizes in the body-centered tetragonal structure, where Ge occupies the 4*b* Wyckoff position, and Pt occupies two sets of inequivalent sites in the 4*a* and 8*h* 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 **n** on the seven AI basis vectors *k* path.

While from SI alone we can ascertain that Pt_{3}Ge 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 Pt_{3}Ge 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 Pt_{3}Ge, we find that the enriched inversion invariant κ_{1} 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 *D*_{4h}), 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 *S*_{4} 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 *C*_{4z}-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.

## ℤ_{12} NONTRIVIAL TCI: GRAPHITE

We also searched the materials with *P*6_{3}/*mmc*, whose point group is *D*_{6h}) 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 _{12}. 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 *κ*_{1} mod 4 is zero. We then calculate the (001) mirror Chern number at *k*_{z} = 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 *k*_{z} = 0. For another mirror symmetric plane, *i* mirror eigenvalue

To ascertain graphite’s nontrivial topology, we then calculate the (*31*, *32*). It can also have glide- and rotation-protected surface states as dictated by the nonvanishing *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.

## THE OTHER DISCOVERED TCIS

### Weak TI coexisting with TCI in PbPt_{3} (S G 221 ) and Au_{4}Ti (S G 87 )

The above two TC materials both have vanishing inversion and weak topological invariants. We also discover two materials, i.e., PbPt_{3} (ICSD[14] 648399) in _{4}Ti (ICSD[14] 109132) in *15*) *ν*_{i} = 1 for *i* = 1, 2, 3 (*ν*_{0} is vanishing for both cases); however, they have different inversion topological invariants, i.e., κ_{1} mod 4 (*31*, *32*) is equal to 0 or 2, respectively.

PbPt_{3} 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 _{4} × ℤ_{8}. 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 (*k*_{z} = 0 or *31*, *32*) and find that they are both equal to −1. This implies that the screw invariant of *32*), is 1. Note that, as

Au_{4}Ti crystallizes in *I*4/*m*), where Au and Ti occupy 8*h* and 2*a* 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 *κ*_{1} mod 4 (*31*, *32*) are both vanishing, while *ν*_{1} = *ν*_{2} = *ν*_{3} = 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 *31*, *32*). It can also host hinge states along the (001) direction, which are protected by the (nonessential) screws *32*).

### TCI Ti_{2}Sn in S G 194

Ti_{2}Sn (ICSD[14] 182428) within _{1} mod 4 (*31*, *32*) is 2, while the strong and weak topological invariants (*15*) ν_{0,1,2,3} are all vanishing. From first-principles calculation, we find that the mirror Chern number for the (*31*, *32*). *C*_{2} around (010) can also protect surface Dirac cones (*31*, *32*). Besides, inversion and screw *31*, *32*).

## TOPOLOGICAL SEMIMETALS

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 Ag*X*F_{3} (*X* = Rb, Cs; *P* and *N*) and AgAs*X* (*X* = Sr, Ba; *X*F_{3} 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 AgAs*X* family, in the high-symmetry line Γ-*A*, the Δ_{7} and Δ_{9} bands cross each other, resulting in a Dirac point protected by *C*_{6v}. It is worth pointing out that for AgAs*X*, the Fermi level exactly threads the Dirac point.

## CONCLUSIONS AND PERSPECTIVES

In this work, on the basis of our newly developed algorithm (*30*), we search for topological materials indicated by ℤ_{8} and ℤ_{12} strong factors in the SI groups. Focusing on **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

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 _{2}. 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.

## METHODS

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 MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/3/eaau8725/DC1

Section S1. First-principles calculated parities

Section S2. *S*_{4} 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 κ_{4} for body-centered lattice.

Table S3. Ab initio calculated κ_{4} for primitive lattice.

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

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

Fig. S1. TB fitting of Pt_{3}Ge 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.

## REFERENCES AND NOTES

**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 https://datadryad.org/resource/doi:10.5061/dryad.8c3hr65. Additional data related to this paper may be requested from the authors.

- Copyright © 2019 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).