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Topological quantum chemistry

Nature volume 547, pages 298305 (20 July 2017) | Download Citation

Abstract

Since the discovery of topological insulators and semimetals, there has been much research into predicting and experimentally discovering distinct classes of these materials, in which the topology of electronic states leads to robust surface states and electromagnetic responses. This apparent success, however, masks a fundamental shortcoming: topological insulators represent only a few hundred of the 200,000 stoichiometric compounds in material databases. However, it is unclear whether this low number is indicative of the esoteric nature of topological insulators or of a fundamental problem with the current approaches to finding them. Here we propose a complete electronic band theory, which builds on the conventional band theory of electrons, highlighting the link between the topology and local chemical bonding. This theory of topological quantum chemistry provides a description of the universal (across materials), global properties of all possible band structures and (weakly correlated) materials, consisting of a graph-theoretic description of momentum (reciprocal) space and a complementary group-theoretic description in real space. For all 230 crystal symmetry groups, we classify the possible band structures that arise from local atomic orbitals, and show which are topologically non-trivial. Our electronic band theory sheds new light on known topological insulators, and can be used to predict many more.

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Acknowledgements

B.B. thanks I. Souza, R. Martin and I. Momennejad for discussions. M.G.V. thanks G. Lopez-Garmendia for help with computational work. B.B., J.C., Z.W. and B.A.B. acknowledge the hospitality of the Donostia International Physics Center, where parts of this work were carried out. J.C. also acknowledges the hospitality of the Kavli Institute for Theoretical Physics, and B.A.B. the hospitality and support of the École Normale Supérieure and Laboratoire de Physique Théorique et Hautes Energies. The work of M.V.G. was supported by the FIS2016-75862-P and FIS2013-48286-C2-1-P national projects of the Spanish MINECO. The work of L.E. and M.I.A. was supported by the Government of the Basque Country (project IT779-13) and the Spanish Ministry of Economy and Competitiveness and FEDER funds (project MAT2015-66441-P). Z.W. and B.A.B., and part of the development of the initial theory and further ab initio work, were supported by the Department of Energy de-sc0016239, a Simons Investigator Award, the Packard Foundation and the Schmidt Fund for Innovative Research. The development of the practical part of the theory, the tables, some of the code development and the ab initio work was funded by NSF EAGER grant number DMR-1643312, ONR N00014-14-1-0330 and NSF-MRSEC DMR-1420541.

Author information

Author notes

    • Barry Bradlyn
    • , L. Elcoro
    • , Jennifer Cano
    • , M. G. Vergniory
    •  & Zhijun Wang

    These authors contributed equally to this work.

Affiliations

  1. Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA

    • Barry Bradlyn
    •  & Jennifer Cano
  2. Department of Condensed Matter Physics, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain

    • L. Elcoro
    •  & M. I. Aroyo
  3. Donostia International Physics Center, P. Manuel de Lardizabal 4, 20018 Donostia-San Sebastián, Spain

    • M. G. Vergniory
    •  & B. Andrei Bernevig
  4. Department of Applied Physics II, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain

    • M. G. Vergniory
  5. Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany

    • M. G. Vergniory
  6. Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

    • Zhijun Wang
    •  & B. Andrei Bernevig
  7. Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany

    • C. Felser
  8. Laboratoire Pierre Aigrain, Ecole Normale Supérieure-PSL Research University, CNRS, Université Pierre et Marie Curie-Sorbonne Universités, Université Paris Diderot-Sorbonne Paris Cité, 24 rue Lhomond, 75231 Paris Cedex 05, France

    • B. Andrei Bernevig
  9. Sorbonne Universités, UPMC Université Paris 06, UMR 7589, LPTHE, F-75005 Paris, France

    • B. Andrei Bernevig

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Contributions

B.B., J.C., Z.W. and B.A.B. provided the theoretical analysis, with input from C.F. J.C. developed specific models to test the theory. L.E. and M.I.A. performed the computerized group-theoretic computations. B.B., L.E. and M.G.V. devised and developed the graph algorithms and the EBR connectivities; L.E. and M.G.V. performed the computerized graph-theory computations. Z.W. discovered the new materials presented here with input from C.F., and performed all of the first-principles calculations.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to B. Andrei Bernevig.

Reviewer Information Nature thanks A. Akhmerov and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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    Supplementary Information

    This file contains Supplementary Text and Data 1-6, Supplementary Figures 1-18, Supplementary Tables 1-16, and additional references.

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DOI

https://doi.org/10.1038/nature23268

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