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General construction and topological classification of crystalline flat bands


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.

Data availability

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

Code availability

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


  1. Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. USA 108, 12233–12237 (2011).

    Article  ADS  Google Scholar 

  2. Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).

    Article  ADS  Google Scholar 

  3. Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).

    Article  ADS  Google Scholar 

  4. Po, H. C., Zou, L., Vishwanath, A. & Senthil, T. Origin of mott insulating behavior and superconductivity in twisted bilayer graphene. Phys. Rev. X 8, 031089 (2018).

    Google Scholar 

  5. Po, H. C., Zou, L., Senthil, T. & Vishwanath, A. Faithful tight-binding models and fragile topology of magic-angle bilayer graphene. Phys. Rev. B 99, 195455 (2019).

    Article  ADS  Google Scholar 

  6. Song, Z. et al. All magic angles in twisted bilayer graphene are topological. Phys. Rev. Lett. 123, 036401 (2019).

    Article  ADS  Google Scholar 

  7. Xie, F., Song, Z., Lian, B. & Bernevig, B. A. Topology-bounded superfluid weight in twisted bilayer graphene. Phys. Rev. Lett. 124, 167002 (2020).

    Article  ADS  Google Scholar 

  8. Mielke, A. Ferromagnetic ground states for the Hubbard model on line graphs. J. Phys. A 24, L73–L77 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  9. Mielke, A. & Tasaki, H. Ferromagnetism in the Hubbard model. Commun. Math. Phys. 158, 341–371 (1993).

    Article  ADS  MATH  Google Scholar 

  10. Wu, C., Bergman, D., Balents, L. & Das Sarma, S. Flat bands and Wigner crystallization in the honeycomb optical lattice. Phys. Rev. Lett. 99, 070401 (2007).

    Article  ADS  Google Scholar 

  11. Huber, S. D. & Altman, E. Bose condensation in flat bands. Phys. Rev. B 82, 184502 (2010).

    Article  ADS  Google Scholar 

  12. Goda, M., Nishino, S. & Matsuda, H. Inverse Anderson transition caused by flatbands. Phys. Rev. Lett. 96, 126401 (2006).

    Article  ADS  Google Scholar 

  13. Chalker, J. T., Pickles, T. S. & Shukla, P. Anderson localization in tight-binding models with flat bands. Phys. Rev. B 82, 104209 (2010).

    Article  ADS  Google Scholar 

  14. Mielke, A. Ferromagnetism in the Hubbard model on line graphs and further considerations. J. Phys. A 24, 3311–3321 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  15. Mielke, A. Exact ground states for the Hubbard model on the Kagome lattice. J. Phys. A 25, 4335–4345 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  16. Mielke, A. Exact results for the U = infinity Hubbard model. J. Phys. A 25, 6507–6515 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  17. Huda, M. N., Kezilebieke, S. & Liljeroth, P. Designer flat bands in quasi-one-dimensional atomic lattices. Phys. Rev. Res. 2, 043426 (2020).

    Article  Google Scholar 

  18. Liu, H. et al. Observation of flat bands due to band hybridization in the 3d-electron heavy-fermion compound CaCu3Ru4O12. Phys. Rev. B 102, 035111 (2020).

    Article  ADS  Google Scholar 

  19. Meier, W. R. et al. Flat bands in the CoSn-type compounds. Phys. Rev. B 102, 075148 (2020).

    Article  ADS  Google Scholar 

  20. Baboux, F. et al. Bosonic condensation and disorder-induced localization in a flat band. Phys. Rev. Lett. 116, 066402 (2016).

    Article  ADS  Google Scholar 

  21. Kollár, A. J., Fitzpatrick, M. & Houck, A. A. Hyperbolic lattices in circuit quantum electrodynamics. Nature 571, 45–50 (2019).

    Article  ADS  Google Scholar 

  22. Bergman, D. L., Wu, C. & Balents, L. Band touching from real-space topology in frustrated hopping models. Phys. Rev. B 78, 125104 (2008).

    Article  ADS  Google Scholar 

  23. Chiu, C. S., Ma, D.-S., Song, Z.-D., Bernevig, B. A. & Houck, A. A. Fragile topology in line-graph lattices with two, three, or four gapped flat bands. Phys. Rev. Res. 2, 043414 (2020).

    Article  Google Scholar 

  24. Ma, D.-S. et al. Spin-orbit-induced topological flat bands in line and split graphs of bipartite lattices. Phys. Rev. Lett. 125, 266403 (2020).

    Article  ADS  Google Scholar 

  25. Lieb, E. H. Two theorems on the Hubbard model. Phys. Rev. Lett. 62, 1201–1204 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  26. Weaire, D. & Thorpe, M. F. Electronic properties of an amorphous solid. I. a simple tight-binding theory. Phys. Rev. B 4, 2508–2520 (1971).

    Article  ADS  Google Scholar 

  27. Tasaki, H. Ferromagnetism in the Hubbard models with degenerate single-electron ground states. Phys. Rev. Lett. 69, 1608–1611 (1992).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Green, D., Santos, L. & Chamon, C. Isolated flat bands and spin-1 conical bands in two-dimensional lattices. Phys. Rev. B 82, 075104 (2010).

    Article  ADS  Google Scholar 

  29. Hatsugai, Y. & Maruyama, I. ZQ topological invariants for polyacetylene, kagome and pyrochlore lattices. Europhys. Lett. 95, 20003 (2011).

    Article  ADS  Google Scholar 

  30. Raoux, A., Morigi, M., Fuchs, J.-N., Piéchon, F. & Montambaux, G. From dia- to paramagnetic orbital susceptibility of massless fermions. Phys. Rev. Lett. 112, 026402 (2014).

    Article  ADS  Google Scholar 

  31. Derzhko, O., Richter, J. & Maksymenko, M. Strongly correlated flat-band systems: the route from Heisenberg spins to Hubbard electrons. Int. J. Mod. Phys. B 29, 1530007 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Leykam, D., Andreanov, A. & Flach, S. Artificial flat band systems: from lattice models to experiments. Adv. Phys. X 3, 1473052 (2018).

    Google Scholar 

  33. Liu, H., Sethi, G., Meng, S. & Liu, F. Orbital design of flat bands in non-line-graph lattices via line-graph wavefunctions. Preprint at (2021).

  34. Regnault, N. et al. Catalogue of flat band stoichiometric materials. Preprint at (2021).

  35. Bradlyn, B. et al. Topological quantum chemistry. Nature 547, 298–305 (2017).

    Article  ADS  Google Scholar 

  36. Elcoro, L. et al. Magnetic topological quantum chemistry. Nat. Commun. 12, 5965 (2021).

    Article  ADS  Google Scholar 

  37. Kruthoff, J., de Boer, J., van Wezel, J., Kane, C. L. & Slager, R.-J. Topological classification of crystalline insulators through band structure combinatorics. Phys. Rev. X 7, 041069 (2017).

    Google Scholar 

  38. Po, H. C., Vishwanath, A. & Watanabe, H. Symmetry-based indicators of band topology in the 230 space groups. Nat. Commun. 8, 50 (2017).

    Article  ADS  Google Scholar 

  39. Watanabe, H., Po, H. C. & Vishwanath, A. Structure and topology of band structures in the 1651 magnetic space groups. Sci. Adv. 4, eaat8685 (2018).

    Article  ADS  Google Scholar 

  40. Hwang, Y., Rhim, J.-W. & Yang, B.-J. Flat bands with band crossings enforced by symmetry representation. Phys. Rev. B 104, L081104 (2021).

    Article  ADS  Google Scholar 

  41. Cano, J. et al. Topology of disconnected elementary band representations. Phys. Rev. Lett. 120, 266401 (2018).

    Article  ADS  Google Scholar 

  42. Po, H. C., Watanabe, H. & Vishwanath, A. Fragile topology and wannier obstructions. Phys. Rev. Lett. 121, 126402 (2018).

    Article  ADS  Google Scholar 

  43. Song, Z.-D., Elcoro, L., Xu, Y.-F., Regnault, N. & Bernevig, B. A. Fragile phases as affine monoids: classification and material examples. Phys. Rev. X 10, 031001 (2020).

    Google Scholar 

  44. Song, Z.-D., Elcoro, L. & Bernevig, B. A. Twisted bulk-boundary correspondence of fragile topology. Science 367, 794–797 (2020).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Vergniory, M. G. et al. Graph theory data for topological quantum chemistry. Phys. Rev. E 96, 023310 (2017).

    Article  ADS  Google Scholar 

  46. Elcoro, L. et al. Double crystallographic groups and their representations on the Bilbao Crystallographic Server. J. Appl. Crystallogr. 50, 1457–1477 (2017).

    Article  Google Scholar 

  47. Xu, Y. et al. High-throughput calculations of magnetic topological materials. Nature 586, 702–707 (2020).

    Article  ADS  Google Scholar 

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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.

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Authors and Affiliations



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).

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