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# Catalogue of topological electronic materials

## Abstract

Topological electronic materials such as bismuth selenide, tantalum arsenide and sodium bismuthide show unconventional linear response in the bulk, as well as anomalous gapless states at their boundaries. They are of both fundamental and applied interest, with the potential for use in high-performance electronics and quantum computing. But their detection has so far been hindered by the difficulty of calculating topological invariant properties (or topological nodes), which requires both experience with materials and expertise with advanced theoretical tools. Here we introduce an effective, efficient and fully automated algorithm that diagnoses the nontrivial band topology in a large fraction of nonmagnetic materials. Our algorithm is based on recently developed exhaustive mappings between the symmetry representations of occupied bands and topological invariants. We sweep through a total of 39,519 materials available in a crystal database, and find that as many as 8,056 of them are topologically nontrivial. All results are available and searchable in a database with an interactive user interface.

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## Data availability

All results are available and searchable with an interactive user interface at http://materiae.iphy.ac.cn. Codes for obtaining the irreducible representations are available from the corresponding author upon reasonable request.

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## References

1. 1.

Kane, C. L. & Mele, E. J. Z. Z 2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).

2. 2.

Bernevig, B. A., Hughes, T. L. & Zhang, S. C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).

3. 3.

Fu, L., Kane, C. L. & Mele, E. J. Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007).

4. 4.

Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045 (2010).

5. 5.

Qi, X. L. & Zhang, S. C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057 (2011).

6. 6.

Schnyder, A. P., Ryu, S., Furusaki, A. & Ludwig, A. W. Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008).

7. 7.

Kitaev, A. Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22–30 (2009).

8. 8.

Chiu, C. K., Teo, J. C., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).

9. 9.

Hsieh, T. H. et al. Topological crystalline insulators in the SnTe material class. Nat. Commun. 3, 982 (2012); corrigendum 4, 1901 (2013).

10. 10.

Song, Z., Fang, Z. & Fang, C. (d-2)-dimensional edge states of rotation symmetry protected topological states. Phys. Rev. Lett. 119, 246402 (2017).

11. 11.

Fang, C. & Fu, L. Rotation anomaly and topological crystalline insulators. Preprint at https://arxiv.org/abs/1709.01929 (2017).

12. 12.

Schindler, F. et al. Higher-order topological insulators. Sci. Adv. 4, eaat0346 (2018).

13. 13.

Wang, Z., Alexandradinata, A., Cava, R. J. & Bernevig, B. A. Hourglass fermions. Nature 532, 189–194 (2016).

14. 14.

Wieder, B. J. et al. Wallpaper fermions and the nonsymmorphic Dirac insulator. Science 361, 246–251 (2018).

15. 15.

Fang, Z. et al. The anomalous Hall effect and magnetic monopoles in momentum space. Science 302, 92–95 (2003).

16. 16.

Murakami, S. Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase. New J. Phys. 9, 356 (2007); corrigendum 10, 029802 (2008).

17. 17.

Wan, X., Turner, A. M., Vishwanath, A. & Savrasov, S. Y. Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011).

18. 18.

Young, S. M. et al. Dirac semimetal in three dimensions. Phys. Rev. Lett. 108, 140405 (2012).

19. 19.

Wang, Z. et al. Dirac semimetal and topological phase transitions in A3Bi (A= Na, K, Rb). Phys. Rev. B 85, 195320 (2012).

20. 20.

Burkov, A. A., Hook, M. D. & Balents, L. Topological nodal semimetals. Phys. Rev. B 84, 235126 (2011).

21. 21.

Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90, 015001 (2018).

22. 22.

Yu, R., Qi, X. L., Bernevig, A., Fang, Z. & Dai, X. Equivalent expression of Z2 topological invariant for band insulators using the non-abelian Berry connection. Phys. Rev. B 84, 075119 (2011).

23. 23.

Fang, C. & Fu, L. New classes of three-dimensional topological crystalline insulators: nonsymmorphic and magnetic. Phys. Rev. B 91, 161105 (2015).

24. 24.

Shiozaki, K., Sato, M. & Gomi, K. Z. Z 2 topology in nonsymmorphic crystalline insulators: Möbius twist in surface states. Phys. Rev. B 91, 155120 (2015).

25. 25.

Bradlyn, B. et al. Topological quantum chemistry. Nature 547, 298–305 (2017).

26. 26.

Po, H. C., Vishwanath, A. & Watanabe, H. Symmetry-based indicators of band topology in the 230 space groups. Nat. Commun. 8, 50 (2017); erratum 8, 931 (2017).

27. 27.

Song, Z., Zhang, T., Fang, Z. & Fang, C. Quantitative mappings between symmetry and topology in solids. Nat. Commun. 9, 3530 (2018).

28. 28.

Khalaf, E., Po, H. C., Vishwanath, A. & Watanabe, H. Symmetry indicators and anomalous surface states of topological crystalline insulators. Phys. Rev. X 8, 031070 (2018).

29. 29.

Kruthoff, J., de Boer, J., van Wezel, J., Kane, C. L. & Slager, R. J. Topological classification of crystalline insulators through band structure combinatorics. Phys. Rev. X 7, 041069 (2017).

30. 30.

Song, Z., Zhang, T. & Fang, C. Diagnosis for nonmagnetic topological semimetals in the absence of spin-orbital coupling. Phys. Rev. X 8, 031069 (2018).

31. 31.

Jain, A. et al. Commentary: the materials project: a materials genome approach to accelerating materials innovation. APL Mater. 1, 011002 (2013).

32. 32.

Hellenbrandt, M. The inorganic crystal structure database (ICSD)—present and future. Crystallogr. Rev. 10, 17–22 (2004).

33. 33.

Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169 (1996).

34. 34.

Elcoro, L. et al. Double crystallographic groups and their representations on the Bilbao Crystallographic Server. J. Appl. Cryst. 50, 1457–1477 (2017).

35. 35.

Chang, G. et al. Unconventional chiral fermions and large topological Fermi arcs in RhSi. Phys. Rev. Lett. 119, 206401 (2017).

36. 36.

Flicker, F. et al. Chiral optical response of multifold fermions. Phys. Rev. B 98, 155145 (2018).

37. 37.

Bradlyn, B. et al. Beyond Dirac and Weyl fermions: unconventional quasiparticles in conventional crystals. Science 353, aaf5037 (2016).

38. 38.

Fang, C., Chen, Y., Kee, H. Y. & Fu, L. Topological nodal line semimetals with and without spin-orbital coupling. Phys. Rev. B 92, 081201 (2015).

39. 39.

Fu, L. & Kane, C. L. Topological insulators with inversion symmetry. Phys. Rev. B 76, 045302 (2007).

40. 40.

Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017).

41. 41.

Langbehn, J., Peng, Y., Trifunovic, L., von Oppen, F. & Brouwer, P. W. Reflection-symmetric second-order topological insulators and superconductors. Phys. Rev. Lett. 119, 246401 (2017).

42. 42.

Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators. Phys. Rev. B 96, 245115 (2017).

43. 43.

Zhang, P. et al. Observation of topological superconductivity on the surface of an iron-based superconductor. Science 360, 182–186 (2018).

44. 44.

Wang, D. et al. Evidence for Majorana bound states in an iron-based superconductor. Science 362, 333–335 (2018).

45. 45.

Yin, J. X. et al. Observation of a robust zero-energy bound state in iron-based superconductor Fe (Te, Se). Nat. Phys. 11, 543–546 (2015).

46. 46.

Zhang, H. et al. Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface. Nat. Phys. 5, 438–442 (2009).

47. 47.

Qian, X., Liu, J., Fu, L. & Li, J. Quantum spin Hall effect in two-dimensional transition metal dichalcogenides. Science 346, 1344–1347 (2014).

48. 48.

Chang, K. et al. Discovery of robust in-plane ferroelectricity in atomic-thick SnTe. Science 353, 274–278 (2016).

49. 49.

Chang, C. Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013).

50. 50.

Gu, Z. C. & Wen, X. G. Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order. Phys. Rev. B 80, 155131 (2009).

51. 51.

Affleck, I. Quantum spin chains and the Haldane gap. J. Phys. Condens. Matter 1, 3047 (1989).

52. 52.

Dzero, M., Sun, K., Galitski, V. & Coleman, P. Topological kondo insulators. Phys. Rev. Lett. 104, 106408 (2010).

53. 53.

Lu, F., Zhao, J., Weng, H., Fang, Z. & Dai, X. Correlated topological insulators with mixed valence. Phys. Rev. Lett. 110, 096401 (2013).

54. 54.

Weng, H., Fang, C., Fang, Z., Bernevig, B. A. & Dai, X. Weyl semimetal phase in noncentrosymmetric transition-metal monophosphides. Phys. Rev. X 5, 011029 (2015).

55. 55.

Huang, S. M. et al. A Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class. Nat. Commun. 6, 7373 (2015).

56. 56.

Ong, S. P. et al. Python Materials Genomics (pymatgen): a robust, open-source python library for materials analysis. Comput. Mater. Sci. 68, 314–319 (2013).

57. 57.

Kresse, G. & Hafner, J. Ab initio molecular-dynamics simulation of the liquid-metal–amorphous-semiconductor transition in germanium. Phys. Rev. B 49, 14251 (1994).

58. 58.

Kresse, G. & Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 6, 15–50 (1996).

59. 59.

Kresse, G. & Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 47, 558 (1993).

60. 60.

Togo, A. & Tanaka, I. First principles phonon calculations in materials science. Scr. Mater. 108, 1 (2015).

61. 61.

Alexandradinata, A., Wang, Z. & Bernevig, B. A. Topological Insulators from Group Cohomology. Phys. Rev. X 6, 021008 (2016).

## Acknowledgements

We are grateful for suggestions and comments from M. Liu, B. Bradlyn, H. Watanabe and B. Wieder. We acknowledge support from the Ministry of Science and Technology of China under grant numbers 2016YFA0302400, 2016YFA0300600 and 2018YFA0305700; the National Science Foundation of China under grant numbers 11674370, 11421092 and 11674369; and the Chinese Academy of Sciences under grant numbers XXH13506-202, XDB07020100 and XDB28000000. We also acknowledge support from the Science Challenge Project (number TZ2016004), the K. C. Wong Education Foundation (GJTD-2018-01), the Beijing Municipal Science and Technology Commission (Z181100004218001) and the Beijing Natural Science Foundation (Z180008).

### Reviewer information

Nature thanks J. Checkelsky, M. Franz and the other anonymous reviewer(s) for their contribution to the peer review of this work.

## Author information

### Author notes

1. These authors contributed equally: Tiantian Zhang, Yi Jiang, Zhida Song

### Affiliations

1. #### Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, China

• Tiantian Zhang
• , Yi Jiang
• , Zhida Song
• , Zhong Fang
• , Hongming Weng
•  & Chen Fang
2. #### University of Chinese Academy of Sciences, Beijing, China

• Tiantian Zhang
• , Yi Jiang
• , Zhida Song
•  & Yuqing He
3. #### Computer Network Information Center, Chinese Academy of Sciences, Beijing, China

• He Huang
•  & Yuqing He
4. #### Kavli Institute for Theoretical Sciences, Chinese Academy of Sciences, Beijing, China

• Zhong Fang
•  & Chen Fang
5. #### Collaborative Innovation Center of Quantum Matter, Beijing, China

• Hongming Weng
6. #### Songshan Lake Materials Laboratory, Dongguan, China

• Hongming Weng
•  & Chen Fang
7. #### CAS Centre for Excellence in Topological Quantum Computation, Beijing, China

• Hongming Weng
•  & Chen Fang
8. #### Physical Science Laboratory, Huairou National Comprehensive Science Center, Beijing, China

• Hongming Weng
•  & Chen Fang

### Contributions

C.F. conceived the work; H.W. and Z.F. were in charge of the numerical methods and checked for consistency with previous works; T.Z. did the major part of the calculations and analyses of materials; Y.J., Z.S., H.H. and Y.H. wrote the code for analysing irreducible representations and symmetry-based indicators; H.H. and Y.H. built the website. C.F., H.W. and Z.F. wrote the main text; and T.Z., Y.J. and Z.S wrote the Methods section and the Supplementary Information.

### Competing interests

The authors declare no competing interests.

### Corresponding authors

Correspondence to Hongming Weng or Chen Fang.

## Extended data figures and tables

1. ### Extended Data Fig. 1 Nodal-ring configuration in BaC20 (nsoc setting).

This material is in space group $${\rm{Pm}}\bar{{\rm{3n}}}$$. a, The three equivalent nodal rings in the $${{\boldsymbol{k}}}_{i}=0\left(i=x,y,z\right)$$ planes, protected by the mirror symmetries on these planes. b, The six equivalent nodal rings in the $${{\boldsymbol{k}}}_{i}\pm {{\boldsymbol{k}}}_{j}=0\left(i,j=x,y,z,\hspace{2.77626pt}i\ne j\right)$$ planes, protected by the glide symmetries on these planes.

2. ### Extended Data Fig. 2 Topological invariants and surface states of Zr(TiH2)2.

a, Brillouin zone for Zr(TiH2)2, in which the yellow plane is $${m}_{1\bar{1}0}$$. b, Wilson loop for Zr(TiH2)2 in the $${m}_{1\bar{1}0}$$ plane. c, One-dimensional helical modes in a cubic Zr(TiH2)2 sample. d, Two-dimensional surface states on each surface of a cubic Zr(TiH2)2 sample.

## Supplementary information

1. ### Supplementary Tables

This file contains five tables. These are the lists of all topological materials theoretically discovered in this work. The materials are sorted into the five classes of “high-symmetry-point semimetals”, “high-symmetry-line semimetals”, “generic-momenta semimetals”, “topological insulators” and “topological crystalline insulators” in Tables I, II, III, IV and V, respectively.

### DOI

https://doi.org/10.1038/s41586-019-0944-6