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

## 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.

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## Data availability

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

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## 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

Authors

### 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.

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.

ac, 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.

ae, Gap extraction from the chemical potential curves at B = 0–6 T. fg, 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.

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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

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

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