## Abstract

Divergent density of states offers an opportunity to explore a wide variety of correlated electron physics. In the thinnest limit, this has been predicted and verified in the ultraflat bands of magic-angle twisted bilayer graphene^{1,2,3,4,5}, the band touching points of few-layer rhombohedral graphite^{6,7,8} and the lightly doped rhombohedral trilayer graphene^{9,10,11}. The simpler and seemingly better understood Bernal bilayer graphene is also susceptible to orbital magnetism at charge neutrality^{7} leading to layer antiferromagnetic states^{12} or quantum anomalous Hall states^{13}. Here we report the observation of a cascade of correlated phases in the vicinity of electric-field-controlled Lifshitz transitions^{14,15} and van Hove singularities^{16} in Bernal bilayer graphene. We provide evidence for the observation of Stoner ferromagnets in the form of half and quarter metals^{10,11}. Furthermore, we identify signatures consistent with a topologically non-trivial Wigner–Hall crystal^{17} at zero magnetic field and its transition to a trivial Wigner crystal, as well as two correlated metals whose behaviour deviates from that of standard Fermi liquids. Our results in this reproducible, tunable, simple system open up new horizons for studying strongly correlated electrons.

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## Data availability

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

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## Acknowledgements

We thank V. I. Fal’ko, L. Levitov, A. H. MacDonald and Di Xiao for discussions. R.T.W. and A.M.S. acknowledge funding from the Center for Nanoscience (CeNS) and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under the SFB 1073 project B10 and under Germany’s Excellence Strategy-EXC-2111-390814868 (MCQST). T.X. and F.Z. acknowledge support from the Army Research Office under grant number W911NF-18-1-0416 and the National Science Foundation under grant numbers DMR-1945351 through the CAREER programme, DMR-2105139 through the CMP programme, and DMR-1921581 through the DMREF programme. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan (grant number JPMXP0112101001) and JSPS KAKENHI (grant numbers 19H05790, 20H00354 and 21H05233).

## Author information

### Authors and Affiliations

### Contributions

A.M.S. fabricated the devices and conducted the measurements and data analysis. K.W. and T.T. grew the hexagonal nitride crystals. T.X. performed the calculations and contributed to the theories. F.Z. supervised the computational and theoretical parts. All authors discussed and interpreted the data. R.T.W. supervised the experiments and the analysis. The manuscript was prepared by A.M.S., F.Z. and R.T.W. with input from all authors.

### Corresponding authors

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## Peer review

### Peer review information

*Nature* thanks Dong-Keun Ki, Folkert de Vries and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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## Extended data figures and tables

### Extended Data Fig. 1 Bilayer graphene devices studied here.

**a**, **b**, Optical images of device A (presented in the main manuscript) (**a**) and device B (**b**). The top hBN is encircled in grey, the upper graphite flake in green, the upper hBN flake in red, the graphite contacts in pink, the bilayer graphene flake in blue, the lower hBN flake in yellow, and the lower graphite flake in purple. **c**, Schematic of the bilayer graphene devices. The colours of different flakes match those in **a**, **b**.

### Extended Data Fig. 2 Stoner physics in the conduction band.

Density derivative of the conductance as a function of the filling factor 𝜈 and the electric field at *B* = 0.8 T for positive filling factors. Two-fold and four-fold LL degeneracies are marked.

### Extended Data Fig. 3 Device Characterizations.

**a**, Fan diagram at *E* = 0 V/nm. The QH states 𝜈 = −4, −2, and 2 are traced as function of the magnetic field and the charge carrier density. **b**, Derivative of the conductance in **a**. **c**, Conductance as a function of the charge carrier density and the electric field at *B* = 2 T. Transitions induced by the electric field are marked by dashed circles. (In **a**–**c**, Integer QH states are labelled by numerals.) **d**, Conductance as a function of the electric field and the magnetic field at 𝜈 = 0. The phase transitions between the canted antiferromagnetic (CAF) and fully layer polarized (FLP) phases are indicated by arrows. **e**, Conductance as a function of the charge carrier density at *E* = 0.08 V/nm and *B* = 2 T (extracted from data in **c**). Here a contact resistance of 7800 Ω was subtracted. **f**, Conductance as a function of the top and bottom-gate voltages at *B* = 0 T in the space of −7 x 10^{11} cm^{−2} < *n* < 7 x 10^{11} cm^{−2} and −0.7 V/nm < *E* < 0.7 V/nm. **g**, Conductance as a function of charge carrier density and magnetic field at *E* = 0.6 V/nm. A contact resistance of *R*_{c} = 2000 Ω + 3000 Ω/T x *B* (T) was subtracted from the measured values. The data are the same measurements presented in Fig 1c of the main manuscript.

### Extended Data Fig. 4 Magnetotransport data of a second device.

Density derivative of the conductance plotted as a function of the charge carrier density and the magnetic field at *E* = −0.8 V/nm for device A (shown in the main manuscript) (**a**) and device B (**b**). The slopes of the lowest integer QH states and of the phases I–IV discussed in the main text are traced by lines in the mirror schematics of the fan diagrams. The corresponding slopes are indicated by arabic numerals. The lines are solid if the states are present and dashed otherwise.

### Extended Data Fig. 5 Zoom-in of phases I–III and A.

**a**, Conductance as a function of charge carrier density and magnetic field at *E* = −0.8 V/nm showing the clear distinction between phases I and A. **b**, Schematic of phases A and I-III at *E* = −0.8 V/nm. **c,d**, Magnetic hysteresis of phase A (**c**)and phase I (**d**). The forward sweeps are shown in blue while the backward ones in red. The hysteresis loop areas are shaded in yellow. **e**, **f**, Conductance as a function of charge carrier density and magnetic field at *E* = −0.6 V/nm (**e**) and *E* = −0.8 V/nm (**f**) showing the clear distinction between phases I, II and III and B and C that show distinct values in conductance and clear steps of conductance at the phase boundaries.

### Extended Data Fig. 6 Additional magnetotransport data at various electric fields.

Conductance and its density derivative plotted as functions of the charge carrier density and the magnetic field at different electric fields.

### Extended Data Fig. 7 Magnetic field hysteresis of phases I–IV.

Hysteresis of the conductance as a function of the out-of-plane magnetic field \({B}_{\perp }\) (**a**, **b**) and the in-plane magnetic field \({B}_{\parallel }\) (**c**) at *E* = −0.6 V/nm and charge carrier densities corresponding to phase I (*n* = −0.85 x 10^{11} cm^{−2}), phase II (*n* = −1.2 x 10^{11} cm^{−2}), phase III (*n* = −1.5 x 10^{11} cm^{−2}), and phase IV (*n* = −2.2 x 10^{11} cm^{−2}), respectively. The forward sweeps are shown in blue while the backward ones in red. The hysteresis loop areas are shaded in yellow. The data shown in **a**, **b** stem from two different sets of measurement. The magnetic field sweeps were started at −1 T and −0.1 T respectively. In (a) the \({B}_{\perp }\) ranges in which phases I–IV are stable are highlighted in green.

### Extended Data Fig. 8 Critical magnetic fields and conductance of phase II at various different electric fields.

**a**–**d**, Critical magnetic fields for devices A and B of phase I (**a**), phase II (**b**), phase III (**c**), and phase IV (**d**) at different electric fields. **e**, Conductance as a function of charge carrier density at different electric fields and *B* = 0. Density regions of stable phase II are highlighted.

### Extended Data Fig. 9 Current dependent measurements.

**a**, **b**, Conductance (**a**) and bias current derivative of conductance (**b**) as a function of bias current *I* and charge carrier density *n* at *E* = −0.7 V/nm and *B* = 0 showing a gap in phases II and III at small currents. **c**, Conductance as a function of *n* at *E* = −0.7 V/nm and *B* = 0 for *I* = 100 nA (blue) and *I* = 1 nA (green). The phases I – IV can not be seen at large currents, indicative of the many-body nature of the phases.

### Extended Data Fig. 10 Temperature dependence of phases I–IV and B–D.

**a**, **b**, Conductance as a function of charge carrier density and temperature *T* at *B* = 0 and *E* = −0.6 V/nm (**a**) and *B* = 0 and *E* = −0.8 V/nm (**b**).**c**, **d**,*R*(**c**) and *R* – *R* (10K) (**d**) as a function of temperature *T* for phase I (*n* = −0.9 x 10^{11} cm^{−2}), phase II (*n* = −1.2 x 10^{11} cm^{−2}), phase III (*n* = −1.5 x 10^{11} cm^{−2}), and phase IV (*n* = −2.2 x 10^{11} cm^{−2}) at *E* = 0.6 V/nm and for the normal state at *E* = 0 V/nm (*n* = −1.0 x 10^{11} cm^{−2}). **e**, *R* – *R* (8K) as a function of temperature *T* for phase I (*n* = −0.9 x 10^{11} cm^{−2}), phase II (*n* = −1.2 x 10^{11} cm^{−2}), phase III (*n* = −1.5 x 10^{11} cm^{−2}), and phase IV (*n* = −2.2 x 10^{11} cm^{−2}) at *E* = −0.6 V/nm and *B* = 0 T and for the phase B (*n* = −2.2 x 10^{11} cm^{−2}), phase C (*n* = −2.5 x 10^{11} cm^{−2}), and phase D (*n* = −4.0 x 10^{11} cm^{−2}) at *E* = −0.6 V/nm and *B* = 0.6 T.

### Extended Data Fig. 11 In-plane magnetic field dependence of phases I–IV.

Conductance as a function of charge carrier density and in-plane magnetic field \({B}_{\parallel }\) at \({B}_{\perp }\)*=* 0 and *E* = −0.6 V/nm. The phase boundary between phase II and III does not shift with increasing the in-plane magnetic field suggesting that both phases likely carry similar in-plane spin order and inter-valley coherence. Compared with phase I, both phases likely have larger magnitudes of spin polarization since they are more stable against large in-plane magnetic fields.

### Extended Data Fig. 12 Experimental indications of a further novel phase close to the Lifshitz transition of the half metal, termed phase V.

We find this additional phase near the density in which the doubly degenerate inner electron pocket is present, consistent with the result r_{s} > 34 (dashed red) for the electron pocket in Fig 1e of the main manuscript. Potentially this phase resembles phase II and/or III but for the doubly degenerate case. **a**, Conductance as a function of charge carrier density for different temperatures at *E* = −0.6 V/nm and *B* = 0.6 T. The insulating correlated phases are highlighted in blue. The maximum/minimum charge carrier density at which phase V is stable is marked by a black square/circle. **b**, Zoom-in of **a** around phase V. **c**, Resistance as a function of temperature *T* for phase V at *E* = −0.6 V/nm, *B* = 0.6 T, and *n* = −3.4 x 10^{11} cm^{−2}. **d**, Conductance as a function of charge carrier density and magnetic field at *E* = −0.6 V/nm. The maximum/minimum charge carrier density at which phase V is stable at *B* = 0.6 T is marked by a black square/circle. **e**, Conductance as a function of charge carrier density and electric field at *B* = 0.6 T. The maximum/minimum charge carrier density at which phase V is stable at *E* = −0.6 V/nm is marked by a black square/circle.

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Seiler, A.M., Geisenhof, F.R., Winterer, F. *et al.* Quantum cascade of correlated phases in trigonally warped bilayer graphene.
*Nature* **608**, 298–302 (2022). https://doi.org/10.1038/s41586-022-04937-1

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DOI: https://doi.org/10.1038/s41586-022-04937-1

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