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General construction and topological classification of crystalline flat bands

Nature Physics volume 18, pages 185–189 (2022)Cite this article

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Abstract

Exotic phases of matter can emerge from the interplay between strong electron interactions and non-trivial topology. Materials that have non-dispersing bands in their electronic band structure, such as twisted bilayer graphene, are prime candidates for strongly interacting physics. However, existing theoretical models for obtaining these ‘flat bands’ in crystals are often too restrictive for experimental realizations. Here we present a generic theoretical technique for constructing perfectly flat bands from bipartite crystalline lattices. Our prescription encapsulates and generalizes the various flat-band models in the literature and is applicable to systems with any orbital content, with or without spin–orbit coupling. Using topological quantum chemistry, we build a complete topological classification in terms of symmetry eigenvalues of all the gapped and gapless flat bands. We also derive criteria for the existence of symmetry-protected band touching points between the flat and dispersive bands, and identify the gapped flat bands as prime candidates for fragile topological phases. Finally, we show that the set of all perfectly flat bands is finitely generated and construct the corresponding bases for all 1,651 Shubnikov space groups.

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Fig. 1: Examples of flat-band constructions in generalized BCLs.

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All data related to this paper are available in the Supplementary Information.

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The code necessary to generate the flat-band bases can be made available upon request from the authors.

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Acknowledgements

We thank M.-R. Li and D.-S. Ma for fruitful discussions and collaboration on related projects. This work is part of a project that has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 101020833). B.A.B. and N.R. were also supported by the US Department of Energy (grant no. DE-SC0016239), and were partially supported by the National Science Foundation (EAGER grant no. DMR 1643312), a Simons Investigator grant (no. 404513), the Office of Naval Research (ONR grant no. N00014-20-1-2303), the Packard Foundation, the Schmidt Fund for Innovative Research, the BSF Israel US foundation (grant no. 2018226), the Gordon and Betty Moore Foundation through grant no. GBMF8685 towards the Princeton theory programme and a Guggenheim Fellowship from the John Simon Guggenheim Memorial Foundation. B.A.B. and N.R. were supported by the NSF-MRSEC (grant bo. DMR-2011750). B.A.B. and N.R. gratefully acknowledge financial support from the Schmidt DataX Fund at Princeton University made possible through a major gift from the Schmidt Futures Foundation. L.E. was supported by the Government of the Basque Country (project IT1301-19) and the Spanish Ministry of Science and Innovation (PID2019-106644GB-I00). Further support was provided by the NSF-MRSEC no. DMR-1420541, BSF Israel US Foundation no. 2018226 and the Princeton Global Network Funds.

Author information

Author notes

  1. These authors contributed equally: Dumitru Călugăru, Aaron Chew and Luis Elcoro.

Authors and Affiliations

  1. Department of Physics, Princeton University, Princeton, NJ, USA

    Dumitru Călugăru, Aaron Chew, Yuanfeng Xu, Nicolas Regnault, Zhi-Da Song & B. Andrei Bernevig

  2. Department of Condensed Matter Physics, University of the Basque Country UPV/EHU, Bilbao, Spain

    Luis Elcoro

  3. Max Planck Institute of Microstructure Physics, Halle, Germany

    Yuanfeng Xu

  4. Laboratoire de Physique de l’École normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Paris, France

    Nicolas Regnault

  5. Donostia International Physics Center, Donostia-San Sebastian, Spain

    B. Andrei Bernevig

  6. IKERBASQUE, Basque Foundation for Science, Bilbao, Spain

    B. Andrei Bernevig

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Contributions

D.C., A.C., L.E. and B.A.B. conceived the work and the main idea of band representation subtraction. D.C., A.C., Z.-D.S., L.E. and B.A.B. contributed to the theory of generalized BCL construction, band representation subtraction and gapless point criteria. D.C. and A.C. analysed the two-dimensional examples of flat-band constructions from Supplementary Section II, with input from Z.-D.S. and L.E. Y.X. performed the first-principles calculations from Supplementary Section IID, and analysed the flat-band crystalline material Ca2Ta2O7. D.C., A.C., L.E. and Z.-D.S. performed the flat-band classification and compiled the tables. All authors discussed the results and wrote the main text and Methods. D.C. and A.C. wrote the Supplementary Information, with input and feedback from L.E., Z.-D.S., N.R., B.A.B. and Y.X.

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Correspondence to B. Andrei Bernevig.

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Peer review information Nature Physics thanks David Carpentier and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Figs. 1–10, discussion (including a pedagogical introduction to BCLs, examples of flat-band constructions and the flat-band classification procedure), and Tables 1–9.

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Călugăru, D., Chew, A., Elcoro, L. et al. General construction and topological classification of crystalline flat bands. Nat. Phys. 18, 185–189 (2022). https://doi.org/10.1038/s41567-021-01445-3

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