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Imaging tunable quantum Hall broken-symmetry orders in graphene


When electrons populate a flat band their kinetic energy becomes negligible, forcing them to organize in exotic many-body states to minimize their Coulomb energy1,2,3,4,5. The zeroth Landau level of graphene under a magnetic field is a particularly interesting strongly interacting flat band because interelectron interactions are predicted to induce a rich variety of broken-symmetry states with distinct topological and lattice-scale orders6,7,8,9,10,11. Evidence for these states stems mostly from indirect transport experiments that suggest that broken-symmetry states are tunable by boosting the Zeeman energy12 or by dielectric screening of the Coulomb interaction13. However, confirming the existence of these ground states requires a direct visualization of their lattice-scale orders14. Here we image three distinct broken-symmetry phases in graphene using scanning tunnelling spectroscopy. We explore the phase diagram by tuning the screening of the Coulomb interaction by a low- or high-dielectric-constant environment, and with a magnetic field. In the unscreened case, we find a Kekulé bond order, consistent with observations of an insulating state undergoing a magnetic-field driven Kosterlitz–Thouless transition15,16. Under dielectric screening, a sublattice-unpolarized ground state13 emerges at low magnetic fields, and transits to a charge-density-wave order with partial sublattice polarization at higher magnetic fields. The Kekulé and charge-density-wave orders furthermore coexist with additional, secondary lattice-scale orders that enrich the phase diagram beyond current theory predictions6,7,8,9,10. This screening-induced tunability of broken-symmetry orders may prove valuable to uncover correlated phases of matter in other quantum materials.

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Fig. 1: Landau level tunnelling spectroscopy in graphene.
Fig. 2: Quantum Hall ferromagnetic gap at charge neutrality.
Fig. 3: Tunable broken-symmetry states of charge-neutral graphene.
Fig. 4: KB order in unscreened charge-neutral graphene.
Fig. 5: CDW order in moderately screened charge-neutral graphene.

Data availability

All data described here are available at Zenodo51.


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We thank F. de Juan, H. Fertig, M. Goerbig, G. Murthy and E. Shimshoni for valuable discussions; B. Kousar for careful reading of the manuscript; D. Dufeu, Ph. Gandit, D. Grand, D. Lepoittevin, J.-F. Motte, P. Plaindoux and L. Veyrat for technical assistance in setting up the experimental system. Samples were prepared at the Nanofab facility of the Néel Institute. This work has received funding from the European Union’s Horizon 2020 research and innovation programme ERC grants QUEST No. 637815 and SUPERGRAPH No. 866365, and the Marie Sklodowska-Curie grant QUESTech No. 766025. A.G.G. acknowledges financial support by the ANR under the grant ANR-18-CE30-0001-01 (TOPODRIVE).

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Authors and Affiliations



A.C. and H.V. fabricated the samples. A.C. and D.W. performed the measurements. A.C., D.W., H.S. and B.S. analysed the data. A.G.G. and C.R. conducted the theoretical analysis. K.W. and T.T. supplied the hBN crystals. F.G. provided technical support on the experiment. C.W. and H.C. contributed to the discussion. B.S. conceived the project and wrote the paper with inputs from all co-authors.

Corresponding author

Correspondence to Benjamin Sacépé.

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The authors declare no competing interests.

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Nature thanks Christopher Gutiérrez 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 Estimation of the dielectric constant of SrTiO3 and rescaling of the gate map.

a, Line cut at Vb = 0 V, averaged on a range of ±20 mV, of the dIt/dVb gate map in Fig. 2a. b, Estimation from the filling factors obtained in a of the charge carrier density n (blue dots), its polynomial fit (blue curve), and computed values of ϵr ϵSTO (red curve), as a function of gate voltage. The fit yields VCNP 13.5 V. c, Rescaling of the gate map of Fig. 2a as a function of ν.

Extended Data Fig. 2 2D-FT decomposition of the asymmetric Kekulé distortion.

a, 3 × 3 nm2 image showing an asymmetric KB pattern, measured at B = 14 T and Vb = 2 mV. b, 2D Fourier transform (2D-FT) of the STM image in a. Yellow circles indicate peaks of the honeycomb lattice, red and blue circles indicate peaks of the bond-density wave. ch, 1.53 × 3 nm2 Filtered images obtained by considering certain peaks of the FFT as indicated in the top right corner of each panel. The Kekulé lattice is drawn in white for reference. The KB order is mostly retrieved by considering only the yellow and red peaks. The asymmetry of the KB pattern is encoded in the blue peaks whose two of them are twice as high as the others due to the K-CDW order. Scale bar is 500 pm.

Extended Data Fig. 3 Contrast inversion and emergence of the Kekulé bond order.

3 × 3 nm2 STM images during which we changed the bias voltage as shown in the bottom insets (the current color bars are tuned separately for each half of the images). a, We start (bottom) at Vb = 32 mV (LL0+) and switch (top) to Vb = −12 mV (LL0−) to observe a contrast inversion of the KB lattice. b, We start (top) at Vb = 200 mV (LL1) and switch (bottom) to Vb = 20 mV (LL0+) and observe the emergence of the KB order from the honeycomb lattice. Scale bars for both images are 500 pm.

Extended Data Fig. 4 Asymmetry reversal of the Kekulé pattern.

3 × 3 nm2 STM images measured at B = 14 T, Vb = 2 mV and at the same position. The three images were measured successively (scanning time : 1 min). A jump occurs in b at the scan line indicated by the red arrows, leading to an inversion of the asymmetry of the Kekulé pattern. The slow scan axis direction is indicated by the blue arrows on the left of each image. Scale bars for the three images are 500 pm.

Extended Data Fig. 5 Change of the Kekulé asymmetry.

10 × 10 nm2 STM images measured at B = 14 T and Vb = 25 mV. In b, the asymmetry pattern changes at the scan line indicated by the red arrows. Scale bars for both images are 1 nm.

Extended Data Fig. 6 Effect of sublattice charge imbalance and a t2 asymmetry on the zeroth Landau level.

a shows that the effect of a finite charge imbalance Δn = nAnB is to gap the zeroth Landau level of graphene. b shows that a hoping asymmetry Δt2 = t2,At2,B also opens up a gap, that depends on momentum k as we move away from the K and K′ points. The parameters are chosen so that Eg is the same on both plots at the K and K′ points, according to Equation (1). Simulations were performed using the kwant software49 for a 41 × 41 hexagonal lattice with ϕ = 0.003 flux per plaquette, in units of the flux quantum. Energies are measured in units of the nearest-neighbor hopping t. For a, Δt2 = 0 and Δn = 0.045, while for b, Δt2 = 0.015 and Δn = 0.

Extended Data Fig. 7 Induced Δt2 asymmetry by interactions.

a shows that a sublattice charge imbalance Δn = nAnB ≠ 0 develops as V1 increases. b shows the concomitant emergence of a second nearest-neighbor bond asymmetry Δt2 = t2At2B ≠ 0, peaking at intermediate values of V1. The simulations are carried out for cylinder circumferences of Ly = 6, 8, 10 sites, all with bond-dimension χ = 1000, using the tenpy package50.

Extended Data Fig. 8 Disappearance of the charge-density wave at low magnetic field in sample AC23.

a, CDW at B = 9 T. b, CDW at B = 7 T. The Moiré superlattice is visible but does not perturb the CDW pattern. c, d, Honeycomb lattice with no CDW at B = 4 T. e, Honeycomb lattice at B = 4 T with residual traces of CDW, see the zoom in f of the white rectangle. Scale bars for all figures are 500 pm.

Extended Data Fig. 9 Charge-density-wave order in sample AC24.

a, Honeycomb lattice at B = 14 T and Vb = −350 mV observed at ν = 0. b, CDW under the same conditions but at Vb = −18 mV. Scale bars for both figures are 500 pm.

Extended Data Table 1 Geometrical parameters of the four measured samples

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This file contains Supplementary Sections 1–6, including Supplementary Figs. 1–6 and references.

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Coissard, A., Wander, D., Vignaud, H. et al. Imaging tunable quantum Hall broken-symmetry orders in graphene. Nature 605, 51–56 (2022).

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