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Topological quantum chemistry

An Author Correction to this article was published on 29 May 2020


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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Figure 1: Schematic of theory applied to graphene with SOC.
Figure 2: Representative band structures for newly predicted materials.
Figure 3: Representative band structures for topologically non-trivial insulators in the Bi1− square net structure type.


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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.

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Authors and Affiliations



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.

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Correspondence to B. Andrei Bernevig.

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

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

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