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A complete catalogue of high-quality topological materials

Naturevolume 566pages480485 (2019) | Download Citation


Using a recently developed formalism called topological quantum chemistry, we perform a high-throughput search of ‘high-quality’ materials (for which the atomic positions and structure have been measured very accurately) in the Inorganic Crystal Structure Database in order to identify new topological phases. We develop codes to compute all characters of all symmetries of 26,938 stoichiometric materials, and find 3,307 topological insulators, 4,078 topological semimetals and no fragile phases. For these 7,385 materials we provide the electronic band structure, including some electronic properties (bandgap and number of electrons), symmetry indicators, and other topological information. Our results show that more than 27 per cent of all materials in nature are topological. We provide an open-source code that checks the topology of any material and allows other researchers to reproduce our results.

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  1. 1.

    Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (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. Topological insulators with inversion symmetry. Phys. Rev. B 76, 045302 (2007).

  4. 4.

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

  5. 5.

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

  6. 6.

    Wang, Z., Weng, H., Wu, Q., Dai, X. & Fang, Z. Three-dimensional Dirac semimetal and quantum transport in Cd3As2. Phys. Rev. B 88, 125427 (2013).

  7. 7.

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

  8. 8.

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

  9. 9.

    Huang, S.-M. et al. An inversion breaking Weyl semimetal state in the TaAs material class. Nat. Commun. 6, 7373 (2015).

  10. 10.

    Lv, B. Q. et al. Experimental discovery of weyl semimetal TaAs. Phys. Rev. X 5, 031013 (2015).

  11. 11.

    Xu, S.-Y. et al. Discovery of a weyl fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015).

  12. 12.

    Bzdušek, T., Wu, Q., Rüegg, A., Sigrist, M. & Soluyanov, A. Nodal-chain metals. Nature 538, 75–78 (2016).

  13. 13.

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

  14. 14.

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

  15. 15.

    Cano, J. et al. Topology of disconnected elementary band representations. Phys. Rev. Lett. 120, 266401 (2018).

  16. 16.

    Elcoro, L. et al. Double crystallographic groups and their representations on the Bilbao Crystallographic Server. J. Appl. Crystallogr. 50, 457–1477 (2017).

  17. 17.

    Vergniory, M. G. et al. Graph theory data for topological quantum chemistry. Phys. Rev. E 96, 023310 (2017).

  18. 18.

    Bradlyn, B. et al. Band connectivity for topological quantum chemistry: band structures as a graph theory problem. Phys. Rev. B 97, 035138 (2018).

  19. 19.

    Band representations of the double space group (BANDREP). Bilbao Crystallographic Server

  20. 20.

    Compatibility relations between representations of the double space groups (DCOMPREL). Bilbao Crystallographic Server

  21. 21.

    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).

  22. 22.

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

  23. 23.

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

  24. 24.

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

  25. 25.

    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).

  26. 26.

    Schindler, F. et al. Higher-order topology in bismuth. Nat. Phys. 14, 918–924 (2018); correction 14, 1067 (2018).

  27. 27.

    Po, H. C., Watanabe, H. & Vishwanath, A. Fragile topology and wannier obstructions. Phys. Rev. Lett. 121, 126402 (2017).

  28. 28.

    Bouhon, A., Black-Schaffer, A. M. & Slager, R.-J. Wilson loop approach to topological crystalline insulators with time reversal symmetry. Preprint at (2018).

  29. 29.

    Song, Z. et al. All “magic angles” are “stable” topological. Preprint at (2018).

  30. 30.

    Bradlyn, B., Wang, Z., Cano, J. & Bernevig, B. A. Disconnected elementary band representations, fragile topology, and Wilson loops as topological indices. Phys. Rev. B 99, 045140 (2019).

  31. 31.

    Song, Z., Zhang, T., Fang, Z. & Fang, C. Quantitative mappings between symmetry and topology in solids. Nat. Commun. 8, 3530 (2018).

  32. 32.

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

  33. 33.

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

  34. 34.

    Kresse, G. & Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 47, 558–561 (1993).

  35. 35.

    Check Topological Mat. Bilbao Crystallographic Server

  36. 36.

    Xu, Q., Yu, R., Fang, Z., Dai, X. & Weng, H. Topological nodal line semimetals in the CaP3 family of materials. Phys. Rev. B 95, 045136 (2017).

  37. 37.

    Fang, C. & Fu, L. Rotation anomaly and topological crystalline insulators. Preprint at (2017).

  38. 38.

    Zhou, X. et al. Topological crystalline insulator states in the Ca2As family. Phys. Rev. B 98, 241104(R) (2018).

  39. 39.

    Tang, F., Po, H. C., Vishwanath, A. & Wan, X. Efficient topological materials discovery using symmetry indicators. Preprint at (2018).

  40. 40.

    Tang, F., Po, H. C., Vishwanath, A. & Wan, X. Topological materials discovery by large-order symmetry indicators. Preprint at (2018).

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We thank B. Bradlyn, J. Cano and M. Aroyo for countless discussions and collaborations, and for their help with the development of the BANDREP section of the BCS, without which none of the present work would have been possible. We thank H. Gross, S. Parkin, U. Schmidt, M. Rampp and the computational resources of the Max Planck Institute at Halle and Garching, as well as the staff at the Atlas supercomputer of the Donostia International Physics Center. We are grateful to H. Lederer and I. Weidl for allowing us access to the Cobra Supercomputer at the Max Planck Gesellschaft (MPG) computing centre. We also thank H. Borrmann of the Max Planck Institute in Dresden for help with the ICSD database. L.E. was supported by the Government of the Basque Country (project IT779-13), the Spanish Ministry of Economy and Competitiveness (MINECO), and the European Fund for Economic and Regional Development (FEDER; project MAT2015-66441-P). M.G.V. was supported by national project IS2016-75862-P of the Spanish MINECO. B.A.B. and Z.W. acknowledge support for the analytical work from the Department of Energy (de-sc0016239), a Simons Investigator Award, the Packard Foundation, and the Schmidt Fund for Innovative Research. Z.W. was also supported by the CAS Pioneer Hundred Talents Program. The computational part of the Princeton work was performed under National Science Foundation (NSF) Early-concept Grants for Exploratory Research (EAGER): DMR 1643312 NOA-AWD1004957, ONR-N00014-14-1-0330, ARO MURI W911NF-12-1-0461, NSF-MRSECDMR-1420541.

Reviewer information

Nature thanks Joseph Checkelsky, Marcel 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: M. G. Vergniory, L. Elcoro


  1. Donostia International Physics Center, San Sebastian, Spain

    • M. G. Vergniory
  2. IKERBASQUE, Basque Foundation for Science, Bilbao, Spain

    • M. G. Vergniory
  3. Applied Physics Department II, University of the Basque Country UPV/EHU, Bilbao, Spain

    • M. G. Vergniory
  4. Department of Condensed Matter Physics, University of the Basque Country UPV/EHU, Bilbao, Spain

    • L. Elcoro
  5. Max Planck Institute for Chemical Physics of Solids, Dresden, Germany

    • Claudia Felser
  6. Laboratoire de Physique de l’École Normale Supérieure, PSL University, CNRS, Sorbonne Université, Université Paris Diderot, Sorbonne Paris Cité, Paris, France

    • Nicolas Regnault
  7. Department of Physics, Princeton University, Princeton, NJ, USA

    • B. Andrei Bernevig
    •  & Zhijun Wang
  8. Physics Department, Freie Universität Berlin, Berlin, Germany

    • B. Andrei Bernevig
  9. Max Planck Institute of Microstructure Physics, Halle, Germany

    • B. Andrei Bernevig
  10. Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing, China

    • Zhijun Wang


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All authors contributed to performing the calculations and analysing the results.

Competing interests

The authors declare no competing interests.

Corresponding authors

Correspondence to B. Andrei Bernevig or Zhijun Wang.

Supplementary information

  1. Supplementary Information

    This file contains Supplementary Text A–U; which includes Supplementary Tables and Figures.

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