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# Interaction-driven band flattening and correlated phases in twisted bilayer graphene

## Abstract

Flat electronic bands, characteristic of ‘magic-angle’ twisted bilayer graphene, host many correlated phenomena1,2,3,4,5,6,7,8,9. Nevertheless, many properties of these bands and emerging symmetry-broken phases are still poorly understood. Here we use scanning tunnelling spectroscopy to examine the evolution of the twisted bilayer graphene bands and related gapped phases as the twist angle between the two graphene layers changes. We detect filling-dependent flattening of the bands that is appreciable even when the angle is well above the magic angle value and so the material is nominally in a weakly correlated regime. Upon approaching the magic angle, we further show that the most prominent correlated gaps begin to emerge when band flattening is maximized around certain integer fillings of electrons per moiré unit cell. Our observations are consistent with a model that suggests that a significant enhancement of the density of states caused by the band flattening triggers a cascade of symmetry-breaking transitions. Finally, we explore the temperature dependence of the cascade and identify gapped features that develop in a broad range of band fillings where superconductivity is expected. Our results highlight the role of interaction-driven band flattening in defining the electronic properties of twisted bilayer graphene.

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## Data availability

The data reported in Figs. 14 can be found on zenodo: https://zenodo.org/record/5173159. Other data that support the findings of this study are available from the corresponding authors on reasonable request.

## Code availability

The code that supports the findings of this study is available from the corresponding authors on reasonable request.

## References

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

2. Kerelsky, A. et al. Maximized electron interactions at the magic angle in twisted bilayer graphene. Nature 572, 95–100 (2019).

3. Choi, Y. et al. Electronic correlations in twisted bilayer graphene near the magic angle. Nat. Phys. 15, 1174–1180 (2019).

4. Xie, Y. et al. Spectroscopic signatures of many-body correlations in magic-angle twisted bilayer graphene. Nature 572, 101–105 (2019).

5. Jiang, Y. et al. Charge order and broken rotational symmetry in magic-angle twisted bilayer graphene. Nature 573, 91–95 (2019).

6. Tomarken, S. L. et al. Electronic compressibility of magic-angle graphene superlattices. Phys. Rev. Lett. 123, 046601 (2019).

7. Nuckolls, K. P. et al. Strongly correlated Chern insulators in magic-angle twisted bilayer graphene. Nature 588, 610–615 (2020).

8. Choi, Y. et al. Correlation-driven topological phases in magic-angle twisted bilayer graphene. Nature 589, 536–541 (2021).

9. Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Flavour Hund’s coupling, Chern gaps and charge diffusivity in moiré graphene. Nature 592, 43–48 (2021).

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

11. Lopes dos Santos, J. M. B., Peres, N. M. R. & Castro Neto, A. H. Graphene bilayer with a twist: electronic structure. Phys. Rev. Lett. 99, 256802 (2007).

12. Uchida, K., Furuya, S., Iwata, J.-I. & Oshiyama, A. Atomic corrugation and electron localization due to moiré patterns in twisted bilayer graphenes. Phys. Rev. B 90, 155451 (2014).

13. Jung, J., Raoux, A., Qiao, Z. & MacDonald, A. H. Ab initio theory of moiré superlattice bands in layered two-dimensional materials. Phys. Rev. B 89, 205414 (2014).

14. Nam, N. N. T. & Koshino, M. Lattice relaxation and energy band modulation in twisted bilayer graphene. Phys. Rev. B 96, 075311 (2017).

15. Carr, S., Fang, S., Zhu, Z. & Kaxiras, E. Exact continuum model for low-energy electronic states of twisted bilayer graphene. Phys. Rev. Res. 1, 013001 (2019).

16. Guinea, F. & Walet, N. R. Continuum models for twisted bilayer graphene: effect of lattice deformation and hopping parameters. Phys. Rev. B 99, 205134 (2019).

17. Bi, Z., Yuan, N. F. Q. & Fu, L. Designing flat bands by strain. Phys. Rev. B 100, 035448 (2019).

18. Parker, D. E., Soejima, T., Hauschild, J., Zaletel, M. P. & Bultinck, N. Strain-induced quantum phase transitions in magic-angle graphene. Phys. Rev. Lett. 127, 027601 (2021).

19. Guinea, F. & Walet, N. R. Electrostatic effects, band distortions, and superconductivity in twisted graphene bilayers. Proc. Natl Acad. Sci. USA 115, 13174–13179 (2018).

20. Goodwin, Z. A. H., Vitale, V., Liang, X., Mostofi, A. A. & Lischner, J. Hartree theory calculations of quasiparticle properties in twisted bilayer graphene. Electron. Struct. 2, 034001 (2020).

21. Cea, T., Walet, N. R. & Guinea, F. Electronic band structure and pinning of Fermi energy to Van Hove singularities in twisted bilayer graphene: a self-consistent approach. Phys. Rev. B 100, 205113 (2019).

22. Cea, T. & Guinea, F. Band structure and insulating states driven by Coulomb interaction in twisted bilayer graphene. Phys. Rev. B 102, 045107 (2020).

23. Rademaker, L., Abanin, D. A. & Mellado, P. Charge smoothening and band flattening due to Hartree corrections in twisted bilayer graphene. Phys. Rev. B 100, 205114 (2019).

24. Klebl, L., Goodwin, Z. A. H., Mostofi, A. A., Kennes, D. M. & Lischner, J. Importance of long-ranged electron–electron interactions for the magnetic phase diagram of twisted bilayer graphene. Phys. Rev. B 103, 195127 (2021).

25. Rademaker, L. & Mellado, P. Charge-transfer insulation in twisted bilayer graphene. Phys. Rev. B 98, 235158 (2018).

26. Carr, S., Fang, S., Po, H. C., Vishwanath, A. & Kaxiras, E. Derivation of Wannier orbitals and minimal-basis tight-binding Hamiltonians for twisted bilayer graphene: first-principles approach. Phys. Rev. Res. 1, 033072 (2019).

27. Calderón, M. J. & Bascones, E. Interactions in the 8-orbital model for twisted bilayer graphene. Phys. Rev. B 102, 155149 (2020).

28. Saito, Y. et al. Hofstadter subband ferromagnetism and symmetry-broken Chern insulators in twisted bilayer graphene. Nat. Phys. 17, 478–481 (2021).

29. Yankowitz, M. et al. Tuning superconductivity in twisted bilayer graphene. Science 363, 1059–1064 (2019).

30. Lu, X. et al. Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene. Nature 574, 653–657 (2019).

31. Arora, H. S. et al. Superconductivity in metallic twisted bilayer graphene stabilized by WSe2. Nature 583, 379–384 (2020).

32. Zondiner, U. et al. Cascade of phase transitions and Dirac revivals in magic-angle graphene. Nature 582, 203–208 (2020).

33. Wong, D. et al. Cascade of electronic transitions in magic-angle twisted bilayer graphene. Nature 582, 198–202 (2020).

34. Koshino, M. et al. Maximally localized wannier orbitals and the extended Hubbard model for twisted bilayer graphene. Phys. Rev. X 8, 031087 (2018).

35. Xie, M. & MacDonald, A. H. Nature of the correlated insulator states in twisted bilayer graphene. Phys. Rev. Lett. 124, 097601 (2020).

36. Xie, M. & MacDonald, A. H. Weak-field Hall resistivity and spin/valley flavor symmetry breaking in MAtBG. Preprint at https://arxiv.org/abs/2010.07928 (2020).

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

38. Hofmann, J. S., Berg, E. & Chowdhury, D. Superconductivity, pseudogap, and phase separation in topological flat bands. Phys. Rev. B 102, 201112 (2020).

39. Bernevig, B. A. et al. Twisted bilayer graphene. V. Exact analytic many-body excitations in Coulomb Hamiltonians: charge gap, Goldstone modes, and absence of Cooper pairing. Phys. Rev. B 103, 205415 (2021).

40. Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene. Nature 590, 249–255 (2021).

41. Hao, Z. et al. Electric field tunable unconventional superconductivity in alternating twist magic-angle trilayer graphene. Science 371, 1133–1138 (2021).

42. Saito, Y. et al. Isospin Pomeranchuk effect in twisted bilayer graphene. Nature 592, 220–224 (2021).

43. Rozen, A. et al. Entropic evidence for a Pomeranchuk effect in magic-angle graphene. Nature 592, 214–219 (2021).

## Acknowledgements

The authors acknowledge discussions with F. Guinea, F. von Oppen, and G. Refael. Funding: This work has been primarily supported by NSF grants DMR-2005129 and DMR-172336; and Army Research Office under Grant Award W911NF17-1-0323. Part of the STM characterization has been supported by NSF CAREER programme (DMR-1753306). Nanofabrication efforts have been in part supported by DOE-QIS programme (DE-SC0019166). S.N.-P. acknowledges support from the Sloan Foundation. J.A. and S.N.-P. also acknowledge support of the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center with support of the Gordon and Betty Moore Foundation through Grant GBMF1250; C.L. acknowledges support from the Gordon and Betty Moore Foundation’s EPiQS Initiative (grant GBMF8682). A.T. and J.A. are grateful for the support of the Walter Burke Institute for Theoretical Physics at Caltech. Y.P. acknowledges support from the startup fund from California State University, Northridge. Y.C. and H.K. acknowledge support from the Kwanjeong Fellowship.

## Author information

Authors

### Contributions

Y.C. and H.K. fabricated samples with the help of R.P. and Y.Z., and performed STM measurements. Y.C., H.K. and S.N.-P. analysed the data. C.L. and Y.P. implemented TBG models. C.L., Y.P. and A.T. provided theoretical analysis of the model results supervised by J.A. S.N.-P. supervised the project. Y.C., H.K., C.L., Y.P., A.T., J.A. and S.N.-P. wrote the manuscript with input from other authors.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

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## Supplementary information

### Supplementary information

Supplementary Discussion and Figs. 1–16.

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Choi, Y., Kim, H., Lewandowski, C. et al. Interaction-driven band flattening and correlated phases in twisted bilayer graphene. Nat. Phys. 17, 1375–1381 (2021). https://doi.org/10.1038/s41567-021-01359-0

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• DOI: https://doi.org/10.1038/s41567-021-01359-0

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