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# Higher-order topology in bismuth

### Subjects

An Author Correction to this article was published on 15 April 2020

An Author Correction to this article was published on 19 September 2018

## 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

• ### 15 April 2020

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

• ### 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). B.A.B. also acknowledges support from a 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

Authors

### 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.

### Corresponding author

Correspondence to Titus Neupert.

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### Competing Interests

The authors declare no competing interests.

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## Supplementary information

### Supplementary Information

Supplementary text, Supplementary Figure S1-S3

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Schindler, F., Wang, Z., Vergniory, M.G. et al. Higher-order topology in bismuth. Nature Phys 14, 918–924 (2018). https://doi.org/10.1038/s41567-018-0224-7

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

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