Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Flavour Hund’s coupling, Chern gaps and charge diffusivity in moiré graphene

Nature volume 592, pages 43–48 (2021)Cite this article

Subjects

Abstract

Interaction-driven spontaneous symmetry breaking lies at the heart of many quantum phases of matter. In moiré systems, broken spin/valley ‘flavour’ symmetry in flat bands underlies the parent state from which correlated and topological ground states ultimately emerge1,2,3,4,5,6,7,8,9,10. However, the microscopic mechanism of such flavour symmetry breaking and its connection to the low-temperature phases are not yet understood. Here we investigate the broken-symmetry many-body ground state of magic-angle twisted bilayer graphene (MATBG) and its nontrivial topology using simultaneous thermodynamic and transport measurements. We directly observe flavour symmetry breaking as pinning of the chemical potential at all integer fillings of the moiré superlattice, demonstrating the importance of flavour Hund’s coupling in the many-body ground state. The topological nature of the underlying flat bands is manifested upon breaking time-reversal symmetry, where we measure energy gaps corresponding to Chern insulator states with Chern numbers 3, 2, 1 at filling factors 1, 2, 3, respectively, consistent with flavour symmetry breaking in the Hofstadter butterfly spectrum of MATBG. Moreover, concurrent measurements of resistivity and chemical potential provide the temperature-dependent charge diffusivity of MATBG in the strange-metal regime11—a quantity previously explored only in ultracold atoms12. Our results bring us one step closer to a unified framework for understanding interactions in the topological bands of MATBG, with and without a magnetic field.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Device structure and demonstration of chemical potential measurement.
Fig. 2: Chemical potential of MATBG as a function of temperature and in-plane magnetic field.
Fig. 3: Probing of the correlated Chern gaps of MATBG in a perpendicular magnetic field.
Fig. 4: Resistivity, electronic compressibility and diffusivity of MATBG in the strange-metal regime.

Similar content being viewed by others

Data availability

The data that support the current study are available from the corresponding authors upon reasonable and well motivated request.

References

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  3. Chen, G. et al. Evidence of a gate-tunable Mott insulator in a trilayer graphene moiré superlattice. Nat. Phys. 15, 237–241 (2019).

    Article  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  6. Sharpe, A. L. et al. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science 365, 605–608 (2019).

    Article  ADS  CAS  Google Scholar 

  7. Serlin, M. et al. Intrinsic quantized anomalous Hall effect in a moiré heterostructure. Science 367, 900–903 (2020).

    Article  ADS  CAS  Google Scholar 

  8. Chen, G. et al. Tunable correlated Chern insulator and ferromagnetism in a moiré superlattice. Nature 579, 56–61 (2020); publisher correction 581, E3 (2020).

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  11. Cao, Y. et al. Strange metal in magic-angle graphene with near Planckian dissipation. Phys. Rev. Lett. 124, 076801 (2020).

    Article  ADS  CAS  Google Scholar 

  12. Brown, P. T. et al. Bad metallic transport in a cold atom Fermi–Hubbard system. Science 363, 379–382 (2019).

    Article  ADS  CAS  Google Scholar 

  13. Lee, P. A., Nagaosa, N. & Wen, X.-G. Doping a Mott insulator: physics of high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).

    Article  ADS  CAS  Google Scholar 

  14. Balents, L. Spin liquids in frustrated magnets. Nature 464, 199–208 (2010).

    Article  ADS  CAS  Google Scholar 

  15. Tanatar, B. & Ceperley, D. M. Ground state of the two-dimensional electron gas. Phys. Rev. B 39, 5005–5016 (1989).

    Article  ADS  CAS  Google Scholar 

  16. Su rez Morell, E., Correa, J. D., Vargas, P., Pacheco, M. & Barticevic, Z. Flat bands in slightly twisted bilayer graphene: tight-binding calculations. Phys. Rev. B 82, 121407 (2010).

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  21. Choi, Y. et al. Electronic correlations in twisted bilayer graphene near the magic angle. Nat. Phys. 15, 1174–1180 (2019). author correction 15, 1205 (2019).

    Article  CAS  Google Scholar 

  22. Kim, S. et al. Direct measurement of the Fermi energy in graphene using a double-layer heterostructure. Phys. Rev. Lett. 108, 116404 (2012).

    Article  ADS  Google Scholar 

  23. Eisenstein, J. P., Pfeiffer, L. N. & West, K. W. Compressibility of the two-dimensional electron gas: measurements of the zero-field exchange energy and fractional quantum Hall gap. Phys. Rev. B 50, 1760–1778 (1994).

    Article  ADS  CAS  Google Scholar 

  24. Setiawan, F. & Das Sarma, S. Temperature-dependent many-body effects in Dirac–Weyl materials: interacting compressibility and quasiparticle velocity. Phys. Rev. B 92, 235103 (2015).

    Article  ADS  Google Scholar 

  25. Padhi, B., Setty, C. & Phillips, P. W. Doped twisted bilayer graphene near magic angles: proximity to Wigner crystallization, not Mott insulation. Nano Lett. 18, 6175–6180 (2018).

    Article  ADS  CAS  Google Scholar 

  26. Wu, S., Zhang, Z., Watanabe, K., Taniguchi, T. & Andrei, E. Y. Chern insulators, van Hove singularities and topological flat bands in magic-angle twisted bilayer graphene. Nat. Mater. https://doi.org/10.1038/s41563-020-00911-2 (2021).

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  29. Saito, Y. et al. Hofstadter subband ferromagnetism and symmetry-broken Chern insulators in twisted bilayer graphene. Nat. Phys. https://doi.org/10.1038/s41567-020-01129-4 (2021).

  30. Das, I. et al. Symmetry-broken Chern insulators and Rashba-like Landau-level crossings in magic-angle bilayer graphene. Nat. Phys. https://doi.org/10.1038/s41567-021-01186-3 (2021).

  31. Streda, P. & Smrcka, L. Thermodynamic derivation of the Hall current and the thermopower in quantising magnetic field. J. Phys. C 16, L895–L899 (1983).

    Article  ADS  CAS  Google Scholar 

  32. Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249 (1976).

    Article  ADS  CAS  Google Scholar 

  33. Georges, A., de’ Medici, L. & Mravlje, J. Strong correlations from Hund’s coupling. Annu. Rev. Condens. Matter Phys. 4, 137–178 (2013).

    Article  ADS  CAS  Google Scholar 

  34. Polshyn, H. et al. Large linear-in-temperature resistivity in twisted bilayer graphene. Nat. Phys. 15, 1011–1016 (2019).

    Article  CAS  Google Scholar 

  35. Bruin, J. N., Sakai, H., Perry, R. S. & Mackenzie, A. P. Similarity of scattering rates in metals showing t-linear resistivity. Science 339, 804–807 (2013).

    Article  ADS  CAS  Google Scholar 

  36. Chowdhury, D., Werman, Y., Berg, E. & Senthil, T. Translationally invariant non-fermi-liquid metals with critical Fermi surfaces: solvable models. Phys. Rev. X 8, 031024 (2018).

    CAS  Google Scholar 

  37. Patel, A. A. & Sachdev, S. Theory of a Planckian metal. Phys. Rev. Lett. 123, 066601 (2019).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  38. Hartnoll, S. A. Theory of universal incoherent metallic transport. Nat. Phys. 11, 54–61 (2015).

    Article  CAS  Google Scholar 

  39. Perepelitsky, E. et al. Transport and optical conductivity in the Hubbard model: a high-temperature expansion perspective. Phys. Rev. B 94, 235115 (2016).

    Article  ADS  Google Scholar 

  40. Pakhira, N. & McKenzie, R. H. Absence of a quantum limit to charge diffusion in bad metals. Phys. Rev. B 91, 075124 (2015).

    Article  ADS  Google Scholar 

  41. Kokalj, J. Bad-metallic behavior of doped Mott insulators. Phys. Rev. B 95, 041110 (2017).

    Article  ADS  Google Scholar 

  42. Efros, A. L. Negative density of states: screening, Einstein relation, and negative diffusion. Phys. Rev. B 78, 155130 (2008).

    Article  ADS  Google Scholar 

  43. Ortix, C., Lorenzana, J. & Di Castro, C. Coulomb-frustrated phase separation phase diagram in systems with short-range negative compressibility. Phys. Rev.Lett. 100, 246402 (2008).

    Article  ADS  CAS  Google Scholar 

Download references

Acknowledgements

We thank S. Das Sarma, A. Georges, F. Guinea, S. Ilani, A. Kapitulnik, L. Levitov, L. Taillefer, S. Todadri, A. Vishwanath, A. Yacoby, D. Bandurin, S. de la Barrera, C. Collignon, A. Fahimniya, D. Rodan-Legrain, Y. Xie and K. Yasuda for discussions. This work has been primarily supported by the US Department of Energy (DOE), Office of Basic Energy Sciences (BES), Division of Materials Sciences and Engineering under award DE-SC0001819 (J.M.P.). Help with transport measurements and data analysis were supported by the National Science Foundation (DMR-1809802) and the STC Center for Integrated Quantum Materials (NSF grant number DMR-1231319) (Y.C.). P.J.-H. acknowledges support from the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF9643. P.J.-H. acknowledges partial support by Fundación Ramon Areces. The development of new nanofabrication and characterization techniques enabling this work has been supported by the US DOE Office of Science, BES, under award DE‐SC0019300. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by MEXT, Japan, grant number JPMXP0112101001, JSPS KAKENHI grant number JP20H00354 and CREST(JPMJCR15F3), JST. This work made use of the Materials Research Science and Engineering Center Shared Experimental Facilities supported by the National Science Foundation (DMR-0819762) and of Harvard’s Center for Nanoscale Systems, supported by the NSF (ECS-0335765).

Author information

Author notes

  1. These authors contributed equally: Jeong Min Park, Yuan Cao

Authors and Affiliations

  1. Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Jeong Min Park, Yuan Cao & Pablo Jarillo-Herrero

  2. Research Center for Functional Materials, National Institute for Materials Science, Tsukuba, Japan

    Kenji Watanabe

  3. International Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Japan

    Takashi Taniguchi

Authors
  1. Jeong Min Park
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuan Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kenji Watanabe
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Takashi Taniguchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Pablo Jarillo-Herrero
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

J.M.P. and Y.C. fabricated the samples and performed transport measurements and numerical simulations. K.W. and T.T. provided hBN samples. J.M.P., Y.C. and P.J.-H. performed data analysis, discussed the results and wrote the manuscript with input from all co-authors.

Corresponding authors

Correspondence to Yuan Cao or Pablo Jarillo-Herrero.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature thanks Philip Phillips, Ke Wang and Shuigang Xu 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.

Extended data figures and tables

Extended Data Fig. 1 Superconductivity and Landau fan diagram of MATBG.

a, Superconducting curves for the ν = −2 – δ and +2 + δ domes of MATBG. The maximum Tc ≈ 2.7 K is determined from the 50% normal resistance of the ν = −2 – δ curve. b, Landau fan diagram of MATBG at 1 K. The CNP shows the main sequence νLL = ±4, ±8,... and the broken-symmetry states νLL = −1, ±2, ±3. There are fans from ν = ±2, where the sequence νLL = +2, +4, +6 and νLL = −2 are seen, respectively. We also find transport evidence of a correlated Chern gap with Chern number C = 3 from ν = +1.

Extended Data Fig. 2 In-plane magnetic-field dependence of μ and Rxx.

a–c, The in-plane magnetic-field dependence of μ and Rxx is shown at ν = −2 (a), ν = −1 (b) and ν = +3 (c). a, b, The hole-doped side features are qualitatively similar to those on the electron-doped side, but weaker. We note that the ‘dip’ in μ on the electron-doped side is analogous to the ‘peak’ on the hole-doped side. For ν = −1, the peak in Rxx and the ‘peak’ feature in μ enhance under B‖. For ν = −2, the peak in Rxx weakens upon applying B‖, whereas the feature in μ does not exhibit a noticeable change. c, For ν = +3, the trend in μ is similar to that for ν = +1, that is, the ‘dip’ feature is enhanced under B‖, although the dependence is generally weaker. There is no noticeable peak in Rxx at ν = +3.

Extended Data Fig. 3 Thermal activation gap analysis and g factors of the correlated states.

a, b, Fitting of temperature-dependent resistance using the Arrhenius formula \({R}_{xx}^{* }\propto \exp [-\varDelta /(2{k}_{{\rm{B}}}T)]\) at ν = +1 (a) and +2 (b) for in-plane magnetic fields of B‖ = 0–11 T. \({R}_{xx}^{* }\) is the background-removed resistance of MATBG. c, B‖ dependence of the thermal activation gap Δ. The extracted g factors are ~0.57, ~1.31 for the ν = +1, +2 states, respectively.

Extended Data Fig. 4 Overlay of inverse compressibility and superconducting dome, and full-range chemical potential data.

a, Temperature dependence of inverse compressibility dμ/dn for T = 2–70 K at B = 0 T. Negative compressibility near ν = +1, +2 persists up to T ≈ 20 K. b, Comparison between dμ/dn and superconducting Tc dome (red points; 20% normal-state resistance) near ν = −2 – δ. The Tc dome occurs near maximum dμ/dn, which is unexpected within a weak-coupling BCS-type mechanism for the superconductivity. c, Same data as in Fig. 2d, but showing the chemical potential beyond ν = ±4.

Extended Data Fig. 5 Perpendicular magnetic-field dependence of Landau level and Chern gaps.

a–e, Gap extraction from the chemical potential curves at B⊥ = 0–6 T. f–g, Magnetic-field dependence of Landau level gaps (f) and correlated Chern gaps (g). Whereas the νLL = ±4, 0 Landau level gaps have an increasing trend with B⊥, the νLL = ±2 gaps show a relatively weak dependence. Similarly, the three correlated Chern gaps also exhibit weak dependence on B⊥. The reason why the Chern gap at ν = 2 + 2ϕ/ϕ0 is larger than the other two might be the difference in their magnetic ground states, with contributions from both orbital and spin degrees of freedom.

Extended Data Fig. 6 In-plane magnetization of MATBG.

a, \({\left(\frac{\partial {M}_{\parallel }}{\partial \nu }\right)}_{T,{B}_{\parallel }}=-{\left(\frac{\partial \mu }{\partial {B}_{\parallel }}\right)}_{T,\nu }\) versus ν. Peaks are visible near νLL = ±1. T = 4 K. b, Magnetization M‖, from integration of the curve in a. M‖ persists near all filling factors ν = ±1, ±2, ±3. The error bands correspond to a confidence level of 95%.

Supplementary information

Supplementary Information

This file contains Supplementary Information, including Supplementary Figures 1–13 and additional references.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Park, J.M., Cao, Y., Watanabe, K. et al. Flavour Hund’s coupling, Chern gaps and charge diffusivity in moiré graphene. Nature 592, 43–48 (2021). https://doi.org/10.1038/s41586-021-03366-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41586-021-03366-w

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing