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Comprehensive search for topological materials using symmetry indicators


Over the past decade, topological materials—in which the topology of electron bands in the bulk material leads to robust, unconventional surface states and electromagnetism—have attracted much attention. Although several theoretically proposed topological materials have been experimentally confirmed, extensive experimental exploration of topological properties, as well as applications in realistic devices, has been restricted by the lack of topological materials in which interference from trivial Fermi surface states is minimized. Here we apply our method of symmetry indicators to all suitable nonmagnetic compounds in all 230 possible space groups. A database search reveals thousands of candidate topological materials, of which we highlight 241 topological insulators and 142 topological crystalline insulators that have either noticeable full bandgaps or a considerable direct gap together with small trivial Fermi pockets. Furthermore, we list 692 topological semimetals that have band crossing points located near the Fermi level. These candidate materials open up the possibility of using topological materials in next-generation electronic devices.

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Fig. 1: Band structures for strong topological insulators.
Fig. 2: Band structures for topological crystalline insulators.
Fig. 3: Band structure for a topological semimetal.

Data availability

All data that support the conclusions of this work can be required from the corresponding author upon reasonable request. All the structures of the topological materials and their electronic energy band plots can be found at


  1. Hasan, M. Z. & Kane, C. L. Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    Article  ADS  CAS  Google Scholar 

  2. Qi, X. L. & Zhang, S. C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    Article  ADS  CAS  Google Scholar 

  3. Fu, L. Topological crystalline insulators. Phys. Rev. Lett. 106, 106802 (2011).

    Article  ADS  Google Scholar 

  4. Slager, R.-J., Mesaros, A., Juricic, V. & Zaanen, J. The space group classification of topological band-insulators. Nat. Phys. 9, 98–102 (2013).

    Article  CAS  Google Scholar 

  5. Ando, Y. & Fu, L. Topological crystalline insulators and topological superconductors: from concepts to materials. Annu. Rev. Condens. Matter Phys. 6, 361–381 (2015).

    Article  ADS  CAS  Google Scholar 

  6. Hsieh, T. H. et al. Topological crystalline insulators in the SnTe material class. Nat. Commun. 3, 982 (2012).

    Article  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

  10. 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).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  12. 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).

    Article  ADS  Google Scholar 

  13. 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).

    Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  19. 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).

    Article  ADS  Google Scholar 

  20. 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).

    Google Scholar 

  21. 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).

    Article  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. 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).

    Article  CAS  Google Scholar 

  26. Xia, Y. et al. Observation of a large-gap topological-insulator class with a single Dirac cone on the surface. Nat. Phys. 5, 398–402 (2009).

    Article  CAS  Google Scholar 

  27. Scanlon, D. O. et al. Controlling bulk conductivity in topological insulators: key role of anti-site defects. Adv. Mater. 24, 2154–2158 (2012).

    Article  CAS  Google Scholar 

  28. Kushwaha, S. K. et al. Sn-doped Bi1.1Sb0.9Te2S bulk crystal topological insulator with excellent properties. Nat. Commun. 7, 11456 (2016).

    Article  ADS  CAS  Google Scholar 

  29. Wang, N. et al. Microscopic origin of the p-type conductivity of the topological crystalline insulator SnTe and the effect of Pb alloying. Phys. Rev. B 89, 045142 (2014).

    Article  ADS  Google Scholar 

  30. 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).

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  32. Watanabe, H., Po, H. C. & Vishwanath, A. Structure and topology of band structures in the 1651 magnetic space groups. Sci. Adv. 4, eaat8685 (2018).

    Article  ADS  Google Scholar 

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

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

  35. Wang, Z., Wieder, B. J., Li, J., Yan, B. & Bernevig, B. A. Higher-order topology, monopole nodal lines, and the origin of large Fermi arcs in transition metal dichalcogenides XTe2 (X=Mo,W). Preprint at (2018).

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

    Article  CAS  Google Scholar 

  37. Kotliar, G. et al. Electronic structure calculations with dynamical mean-field theory. Rev. Mod. Phys. 78, 865–951 (2006).

    Article  ADS  CAS  Google Scholar 

  38. Mravlje, J., Aichhorn, M. & Georges, A. Origin of the high Néel temperature in SrTcO3. Phys. Rev. Lett. 108, 197202 (2012).

    Article  ADS  Google Scholar 

  39. Baumberger, F. et al. Fermi surface and quasiparticle excitations of Sr2RhO4. Phys. Rev. Lett. 96, 246402 (2006).

    Article  ADS  CAS  Google Scholar 

  40. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).

    Article  ADS  CAS  Google Scholar 

  41. Tran, F. & Blaha, P. Accurate band gaps of semiconductors and insulators with a semilocal exchange-correlation potential. Phys. Rev. Lett. 102, 226401 (2009).

    Article  ADS  Google Scholar 

  42. Feng, W., Xiao, D., Zhang, Y. & Yao, Y. Half-Heusler topological insulators: a first-principles study with the Tran-Blaha modified Becke-Johnson density functional. Phys. Rev. B 82, 235121 (2010).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  44. Fang, C., Gilbert, M. J. & Bernevig, B. A. Bulk topological invariants in noninteracting point group symmetric insulators. Phys. Rev. B 86, 115112 (2012).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  46. 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).

    Google Scholar 

  47. Blaha, P., Schwarz, K., Madsen, G. K. H., Kvasnicka, D. & Luitz, J. WIEN2K, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties (Karlheinz Schwarz, Technische Univ. Wien, Austria, 2001).

    Google Scholar 

  48. Koller, D., Tran, F. & Blaha, P. Merits and limits of the modified Becke-Johnson exchange potential. Phys. Rev. B 83, 195134 (2011).

    Article  ADS  Google Scholar 

  49. Singh, D. J. Structure and optical properties of high light output halide scintillators. Phys. Rev. B 82, 155145 (2010).

    Article  ADS  Google Scholar 

  50. Oinuma, H. et al. Three-dimensional band structure of LaSb and CeSb: absence of band inversion. Phys. Rev. B 96, 041120(R) (2017).

    Article  ADS  Google Scholar 

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F.T. and X.W. were supported by the National Key R&D Program of China (grants 2017YFA0303203 and 2018YFA0305704), the National Natural Science Foundation of China (NSFC; grants 11525417, 11834006, 51721001 and 11790311) and the Excellent Programme at Nanjing University. F.T. was also supported by Program B for outstanding PhD candidates of Nanjing University. X.W. was partially supported by a QuantEmX award funded by the Gordon and Betty Moore Foundation’s Emergent Phenomena in Quantum Systems (EPIQS) Initiative through the Institute for Complex Adaptive Matter (ICAM-I2CAM; grant GBMF5305) and by ICAM. A.V. is supported by National Science Foundation (NSF) grant DMR-1411343, and by a Simons Investigator Grant. H.C.P. is supported by a Pappalardo Fellowship at MIT. We also thank G. Yao for technical support with computers and in setting up the website.

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Nature thanks Joseph Checkelsky and Marcel Franz for their contribution to the peer review of this work.

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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.

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Correspondence to Xiangang Wan.

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This file contains Supplementary Information I-VI and References; including Tables I-VIII.

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Tang, F., Po, H.C., Vishwanath, A. et al. Comprehensive search for topological materials using symmetry indicators. Nature 566, 486–489 (2019).

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