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# Correlated Hofstadter spectrum and flavour phase diagram in magic-angle twisted bilayer graphene

## Abstract

In magic-angle twisted bilayer graphene, the moiré superlattice potential gives rise to narrow electronic bands that support a multitude of many-body quantum phases. Further richness arises in the presence of a perpendicular magnetic field, where the interplay between moiré and magnetic length scales leads to fractal Hofstadter subbands. In this strongly correlated Hofstadter platform, multiple experiments have identified gapped topological and correlated states, but little is known about the phase transitions between them in the intervening compressible regimes. Here we simultaneously unveil sequences of broken-symmetry Chern insulators and resolve sharp phase transitions between competing states with different topological quantum numbers and different occupations of the spin-valley flavour. Our measurements determine the energy spectrum of interacting Hofstadter subbands in magic-angle twisted bilayer graphene and map out the phase diagram of flavour occupancy. In addition, we observe full lifting of the degeneracy of the zeroth Landau levels together with level crossings, indicating moiré valley splitting. We propose a unified flavour polarization mechanism to understand the intricate interplay of topology, interactions and symmetry breaking as a function of density and applied magnetic field in this system.

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## Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request. Source data are provided with this paper.

## Code availability

The codes that support the findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

We thank D. Goldhaber-Gordon and O. Vafek for helpful discussions. This work was supported by the QSQM, an Energy Frontier Research Center funded by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under award no. DE-SC0021238. B.E.F. acknowledges a Stanford University Terman Fellowship and an Alfred P. Sloan Foundation Fellowship. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by MEXT, Japan (grant no. JPMXP0112101001) and JSPS KAKENHI (grant no. JP20H00354). S.A.K. acknowledges support from the Department of Energy, Office of Basic Energy Sciences, under contract no. DEAC02-76SF00515. B.A.F. acknowledges a Stanford Graduate Fellowship. M.E.B. acknowledges support from the Marvin Chodorow Postdoctoral Fellowship of the Applied Physics Department, Stanford University. Y.S. was supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative through grants GBMF 4302 and GBMF 8686. P.P. acknowledges partial support from NSF grant DMR-2111379. Part of this work was performed at the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation under award ECCS-2026822.

## Author information

Authors

### Contributions

J.Y., B.A.F. and B.E.F. designed and conducted the scanning SET experiments. M.E.B. and Z.-X.S. designed and conducted the MIM experiments. B.A.F. fabricated the sample. Z.H., Y.S., S.A.K. and P.P. contributed to theoretical analysis. K.W. and T.T. provided hBN crystals. All authors participated in discussions and in writing of the manuscript.

### Corresponding author

Correspondence to Benjamin E. Feldman.

## Ethics declarations

### Competing interests

Z.-X.S. is a co-founder of PrimeNano Inc., which licensed the MIM technology from Stanford University for commercial instruments.

## Peer review

### Peer review information

Nature Physics thanks the anonymous reviewers 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

### Extended Data Fig. 1 Spatial dependence of electronic compressibility at zero field.

Linecut of /dn as a function of gate voltage (Vg) measured along a line in the top left contact area of the device shown in Fig. 1a. The Landau fans shown in Fig. 1 and Extended Data Fig. 4 are taken at x = 40 nm and x = 300 nm, respectively. The sawtooth pattern on the electron side generically exhibits a larger amplitude and negative /dn at the ν = 2 transition. Dashed lines denote gate voltages corresponding to ν = ±4.

### Extended Data Fig. 2 Spatial dependence of zLL and ChIs.

Linecut of /dn at 11 T. The positions with x > 0 correspond to those shown in Extended Data Fig. 1. While the overall strengths of the broken-symmetry zLLs is spatially dependent, those at odd integers have comparable strengths to those at even integers independent of position.

### Extended Data Fig. 3 Comparison of inverse electronic compressibility and MIM measurements.

Selected /dn measurements as a function of moiré filling factor ν at different perpendicular magnetic fields reproduced from Fig. 1d. The curves are vertically spaced according to the applied magnetic field and the grey lines indicate the field dependence of the incompressible states identified in Fig. 1e. b, MIM measurements of the local conductivity at the same sample position as a function of ν and at the same magnetic fields. A decrease in MIM-Im corresponds to a decrease in conductivity confirming the main ChI and zLLs are associated with local resistive behavior. The MIM measurements were taken at T = 450 mK.

### Extended Data Fig. 4 Landau fan of ChIs at a second location.

/dn measured at a second location (blue dot in Fig. 1a) with twist angle θ = 1.09˚ showing qualitatively similar ChIs/zLLs sequences and phase boundaries as in Fig. 1d.

### Extended Data Fig. 5 Landau fan at higher temperature.

/dn measured at the location of Fig. 1d at temperature T = 1.7 K. The ChIs/zLLs and negative /dn features are thermally broadened and weakened, but the qualitative pattern is unchanged. Interestingly, negative /dn features that mark phase transitions appear closer to the adjacent incompressible states.

### Extended Data Fig. 6 Spatial dependence and competing ChIs.

Higher resolution Landau fans in the vicinity of (𝜈 = −8/3, B = 8.7 T) taken in the locations of Fig. 1d and Extended Data Fig. 4. The pattern of dominant ChIs and detailed regions of negative /dn are distinct in each location. c-e, Spatial linecuts (same trajectory as in Extended Data Fig. 1) taken at B = 9.4, 8.7, and 7.4 T, respectively.

### Extended Data Fig. 7 Hofstadter spectrum from a second location.

Correlated Hofstadter spectrum extracted from the data in Extended Data Fig. 4. The spectrum qualitatively agrees with that in Fig. 2c with a slightly smaller total bandwidth.

### Extended Data Fig. 8 Hysteresis as a function of B.

/dn measured as a function of both increasing (red) and decreasing (blue) B at fixed densities. Hysteretic regions are highlighted in yellow. d-e, Hysteresis as a function of density, reproduced from Fig. 3a, d. Overlaid yellow lines indicate ranges over which we observe hysteresis as a function of B. Solid (dashed) lines indicate /dnup/dndown > 0 (<0), respectively. The ranges of ν and B in which hysteresis occurs are independent of which is swept as the fast axis.

### Extended Data Fig. 9 zLL gap evolution at a second location.

Thermodynamic gaps of νC = ±2 zLLs extracted from the data presented in Extended Data Fig. 4.

### Extended Data Fig. 10 Phenomenological model of zLL evolution in the presence of mirror symmetry breaking.

Schematic of the moiré valley splitting Δv between $$\kappa _M,\kappa _M^\prime$$ by Δv. b, Evolution of the zLLs in the presence of inter-flavor Coulomb repulsion U (ignoring single-particle effects). Four zLLs with different flavors are split linearly in B within each moiré valley (red and blue, respectively), resulting in LL crossings. The corresponding Chern numbers are labeled within the gaps. c, B dependence of zLL gaps derived from b. d, Evolution of the zLLs if the Zeeman coupling is different for spin and valley flavors. Such single-particle gaps lead to different effective interactions U and U’ between LLs corresponding to different flavors within each moiré valley. This modifies the fields at which LL crossings occur, reducing B1, B2 and B3 relative to the predicted field $$B_1^\prime$$ at which the gaps at |νC| = 2 close and those at odd integer fillings saturate. e, B dependence of the zLL gaps derived from d.

## Supplementary information

### Supplementary information

Supplementary Sections 1–9, Figs. 1–4, Tables 1 and 2, and References.

## Source data

### Source Data Fig. 1

Data for all line plots.

### Source Data Fig. 3

Data for all line plots.

### Source Data Fig. 4

Data for all line plots.

### Source Data Extended Data Fig. 3

Data for all line plots.

### Source Data Extended Data Fig. 8

Data for all line plots.

### Source Data Extended Data Fig. 9

Data for all line plots.

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Yu, J., Foutty, B.A., Han, Z. et al. Correlated Hofstadter spectrum and flavour phase diagram in magic-angle twisted bilayer graphene. Nat. Phys. 18, 825–831 (2022). https://doi.org/10.1038/s41567-022-01589-w

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• DOI: https://doi.org/10.1038/s41567-022-01589-w