Abstract

The mathematical field of topology has become a framework in which to describe the low-energy electronic structure of crystalline solids. Typical of a bulk insulating three-dimensional topological crystal are conducting two-dimensional surface states. This constitutes the topological bulk–boundary correspondence. Here, we establish that the electronic structure of bismuth, an element consistently described as bulk topologically trivial, is in fact topological and follows a generalized bulk–boundary correspondence of higher-order: not the surfaces of the crystal, but its hinges host topologically protected conducting modes. These hinge modes are protected against localization by time-reversal symmetry locally, and globally by the three-fold rotational symmetry and inversion symmetry of the bismuth crystal. We support our claim theoretically and experimentally. Our theoretical analysis is based on symmetry arguments, topological indices, first-principles calculations, and the recently introduced framework of topological quantum chemistry. We provide supporting evidence from two complementary experimental techniques. With scanning-tunnelling spectroscopy, we probe the signatures of the rotational symmetry of the one-dimensional states located at the step edges of the crystal surface. With Josephson interferometry, we demonstrate their universal topological contribution to the electronic transport. Our work establishes bismuth as a higher-order topological insulator.

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Change history

  • 19 September 2018

    In the version of this Article originally published, B. Andrei Bernevig was missing the following two additional affiliations: Physics Department, Freie Universitat Berlin, Berlin, Germany, and Max Planck Institute of Microstructure Physics, Halle, Germany. This has been corrected in the online versions of the Article.

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Acknowledgements

F.S. and T.N. acknowledge support from the Swiss National Science Foundation (grant number 200021_169061) and from the European Union’s Horizon 2020 research and innovation programme (ERC-StG-Neupert-757867-PARATOP). M.G.V. was supported by the IS2016-75862-P national project of the Spanish MINECO. A.M.C. wishes to thank the Kavli Institute for Theoretical Physics, which is supported by the National Science Foundation under grant number NSF PHY-1125915, for hosting during some stages of this work. A.M., S.S., A.Y.K., R.D., H.B. and S.G. thank M. Houzet, who drew their attention to recently published work on higher-order topological insulators. They were supported by the ANR grants DIRACFORMAG, MAGMA and JETS. A.Y. acknowledges support from NSF-MRSEC programmes through the Princeton Center for Complex Materials DMR-142054, NSF-DMR-1608848 and ARO-MURI programme W911NF-12-1-046. B.A.B. acknowledges support for the analytic work from the Department of Energy (de-sc0016239), the Simons Investigator Award, the Packard Foundation, and the Schmidt Fund for Innovative Research. The computational part of the Princeton work was performed under NSF EAGER grant DMR-1643312, ONR-N00014-14-1-0330, ARO MURI W911NF-12-1-0461 and NSF-MRSEC DMR-1420541.

Author information

Affiliations

  1. Department of Physics, University of Zurich, Zurich, Switzerland

    • Frank Schindler
    • , Ashley M. Cook
    •  & Titus Neupert
  2. Department of Physics, Princeton University, Princeton, NJ, USA

    • Zhijun Wang
  3. Donostia International Physics Center, Donostia-San Sebastian, Spain

    • Maia G. Vergniory
  4. Department of Applied Physics II, Faculty of Science and Technology, University of the Basque Country UPV/EHU, Bilbao, Spain

    • Maia G. Vergniory
  5. IKERBASQUE, Basque Foundation for Science, Bilbao, Spain

    • Maia G. Vergniory
  6. LPS, Univ. Paris-Sud, CNRS, UMR 8502, Paris, France

    • Anil Murani
    • , Alik Yu. Kasumov
    • , Richard Deblock
    • , Hélène Bouchiat
    •  & Sophie Guéron
  7. CSNSM, Univ. Paris-Sud, IN2P3, UMR 8609, Paris, France

    • Shamashis Sengupta
  8. Institute of Microelectronics Technology and High Purity Materials RAS, Chernogolovka, Russia

    • Alik Yu. Kasumov
  9. Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, NJ, USA

    • Sangjun Jeon
    • , Ali Yazdani
    •  & B. Andrei Bernevig
  10. Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, NY, USA

    • Ilya Drozdov
  11. Physics Department, Freie Universitat Berlin, Berlin, Germany

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

    • B. Andrei Bernevig

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Contributions

F.S., A.M.C., B.A.B. and T.N. carried out the theoretical analysis and model calculations. Z.W. and M.G.V. performed the first-principles calculations and topological quantum chemistry analysis. A.M., S.S., A.Y.K., R.D., H.B., and S.G. conceived and carried out the transport experiments including crystal growth. S.J., I.D. and A.Y. conceived and carried out the STM/STS experiments.

Competing Interests

The authors declare no competing interests.

Corresponding author

Correspondence to Titus Neupert.

Supplementary information

  1. Supplementary Information

    Supplementary text, Supplementary Figure S1-S3

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DOI

https://doi.org/10.1038/s41567-018-0224-7

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