Abstract
In a dilute two-dimensional electron gas, Coulomb interactions can stabilize the formation of a Wigner crystal1,2,3. Although Wigner crystals are topologically trivial, it has been predicted that electrons in a partially filled band can break continuous translational symmetry and time-reversal symmetry spontaneously, resulting in a type of topological electron crystal known as an anomalous Hall crystal4,5,6,7,8,9,10,11. Here we report signatures of a generalized version of the anomalous Hall crystal in twisted bilayer–trilayer graphene, whose formation is driven by the moiré potential. The crystal forms at a band filling of one electron per four moiré unit cells (ν = 1/4) and quadruples the unit-cell area, coinciding with an integer quantum anomalous Hall effect. The Chern number of the state is exceptionally tunable, and it can be switched reversibly between +1 and −1 by electric and magnetic fields. Several other topological electronic crystals arise in a modest magnetic field, originating from ν = 1/3, 1/2, 2/3 and 3/2. The quantum geometry of the interaction-modified bands is likely to be very different from that of the original parent band, which enables possible future discoveries of correlation-driven topological phenomena.
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Acknowledgements
We thank L. Fu, L. Balents, M. Franz, M. Zaletel and A. Young for helpful discussions. Experiments at UBC were undertaken with support from the Natural Sciences and Engineering Research Council of Canada; the Canada Foundation for Innovation; the Canadian Institute for Advanced Research; the Max Planck-UBC-UTokyo Centre for Quantum Materials and the Canada First Research Excellence Fund, Quantum Materials and Future Technologies Program; and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant Agreement No. 951541. Work at UW was supported by National Science Foundation (NSF) CAREER award no. DMR-2041972 and NSF MRSEC 2308979. The development of twisted graphene samples was partially supported by the Department of Energy, Basic Energy Science Programs under award DE-SC0023062. D.W. was supported by an appointment to the Intelligence Community Postdoctoral Research Fellowship Program at University of Washington administered by Oak Ridge Institute for Science and Education through an interagency agreement between the US Department of Energy and the Office of the Director of National Intelligence. M.Y. acknowledges support from the State of Washington-funded Clean Energy Institute. K.W. and T.T. acknowledge support from the JSPS KAKENHI (Grant Nos 21H05233 and 23H02052) and World Premier International Research Center Initiative (WPI), MEXT, Japan. Y.-H.Z. was supported by the National Science Foundation under Grant No. DMR-2237031. This work made use of shared fabrication facilities at UW provided by NSF MRSEC 2308979.
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R.S. performed the measurements in the Folk lab at UBC and analysed the data. D.W. made the sample and performed initial transport measurements in the Yankowitz lab at UW. M.Y. and J.F. supervised the measurements. R.S., M.Y. and J.F. wrote the manuscript with B.Z. and Y.Z. providing support with the theory. K.W. and T.T. provided the hBN crystals.
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Extended data figures and tables
Extended Data Fig. 1 Characterization of twist angle homogeneity.
a, Line-cut of ρxx(n) for three adjacent sets of voltage probes along the Hall bar, at fixed D = 0 and T = 4.2 K. Inset: optical micrograph of the device. Electrical contacts are labeled A-C and 1-4. The scale bar is 5 μm. b-d, Gate voltage dependence of ρxx measured from contact pairs B-1, 1-2, and 2-3, respectively. The dashed line shows the contour of D = 0, where line-cuts in a were obtained. The arrows show orthogonal axes of increasing n and D, related to Vtg and Vbg via the linear transformation described in the Methods. Positive D corresponds to electrons pushed to toward the bilayer, so the fully-filled valence band lives more toward the bilayer and the conduction band lives more toward the trilayer. The close alignment of ρxx features for different contact pairs in a demonstrates the high degree of twist angle homogeneity across the length of the sample. The source and drain contacts used for b were A and C. All other measurements used B and C as source and drain contacts.
Extended Data Fig. 2 Temperature dependence of the ν = 1/4 state.
a, Deviation of the Hall plateau from h/e2 quantization, as a function of displacement field and temperature, obtained with a small B⊥ = − 50 mT. Since ρxy < 0, δρxy is shown as h/e2 + ρxy for clarity. Numerical subscripts (1-4 in this case) indicate the contact pairs used for the measurement, as labeled in the inset of Extended Data Fig. 1a. b, ρxx measured in the same way as a. c, Percent deviation of the Hall resistivity, \(\overline{\delta {\rho }_{xy}}=100\times \delta {\rho }_{xy}\times {(h/{e}^{2})}^{-1}\). d, Longitudinal resistivity averaged using data points between D = 526 and 533 mV nm−1,\(\overline{{\rho }_{xx}}\). Error bars represent one standard deviation. e, f, − ρxy(T) and ρxx(T), respectively, at fixed D = 532 mV nm−1 and B⊥ = − 50 mT.
Extended Data Fig. 3 Filling factor and temperature dependence of the ν = 1/4 TEC state.
a, Temperature dependence of ρxx,1−3 as ν is tuned across the TEC state, at fixed D = 531 mV nm−1 with a small magnetic field of B⊥ = − 50 mT applied. b, Line-cuts from a at selected values of T between 0.65 K and 10 mK. c-f, Measurements obtained under the same conditions as a-b but for c-d ρxy,2−4 and e-f ρ1−4 (corresponding to a mixture of longitudinal and Hall geometries since contacts 1 and 4 are separated by ≈ 1 μm along the length of the channel and located by opposite sides of the Hall bar). Contact 2 fails severely for ν ⪅ 0.15, leading to extremely large and rapidly oscillating resistance. Similar contact issues in the TEC state at ν = 1/4 likely also explain the large fluctuating resistance of ρxy,2−4 near base temperature. ρ1−4 exhibits a mixture of ρxx-like and ρxy-like properties when probing metallic states but faithfully reflects the quantization of the anomalous Hall plateau in the topological gap (ν ≈ 0.25) since the ρxx contribution vanishes.
Extended Data Fig. 4 Doping dependence of the anomalous Hall effect.
a, Antisymmetrized ρxy(B⊥) hysteresis loops at increasing values of ν away from ν = 1/4. Data obtained at fixed D = 532 mV nm−1 and T = 300 mK. The solid/dashed traces correspond to forward (↑) and reverse (↓) scans of B⊥, respectively. b, Map of Δρxy/2 = (ρxy(B↑) − ρxy(B↓))/2, showing the AHE over a broad range of ν. The color scale is saturated to show the weakest AHE features near ν = 1. c, Line-cut of Δρxy/2 at B⊥ = 0, showing that the small anomalous Hall effect vanishes near ν = 1.
Extended Data Fig. 5 Additional Landau fans at fixed D.
a, b, Landau fans of antisymmetrized ρxy and symmetrized ρxx, respectively, obtained at D = 570 mV nm−1. Dashed lines correspond to the trajectories of states with Chern number C = +1 as described by the Streda formula, tracing back to ν = 1/2, 2/3 and 3/2 at B⊥ = 0. Deviations in both ρxy and ρxx along these trajectories indicate the formation of TEC states. Additionally, both electron- and hole-like quantum oscillations emerge from the correlated insulators at ν = 1 and ν = 2 and develop into quantum Hall states at high field. At fixed ν = 1 and 2, ρxx is diverging whereas ρxy = 0 and abruptly changes sign with ν. Together, these features indicate that the correlated insulating states have C = 0. Notably, the ν = 1 state only becomes insulating in a small B⊥ and is metallic at B⊥ ≈ 0. This metal-to-insulator transition with B⊥ is seen irrespective of the precise value of D. c, d, Similar Landau fans obtained at D = 531 mV nm−1. Dashed lines correspond to the trajectories of states with Chern numbers C = +1 and +2 tracing back to ν = 3/4 at B⊥ = 0. There are weak deviations in both ρxy and ρxx along these trajectories, suggesting the possibility of gapless TEC states with a four-fold enlargement of the moiré unit cell.
Extended Data Fig. 6 First-order phase transition and additional characterization of the ν = 1/4 and 1/3 states.
a, b, Landau fans of antisymmetrized ρxy and symmetrized ρxx obtained at fixed D = 527 mV nm−1, respectively. (Which is larger than the D = 526 mV nm−1 fans shown in Fig. 2d of the main text). Panel a is overlaid with dashed/dotted lines, indicating ν = 0.322 and B⊥ = 0.5 T. Panel b is overlaid with dashed lines that show the expected trajectories of ∣C∣ = 1 states originating from ν = 1/4 and ν = 1/3. c, ρxy for forward and reverse scans of ν, at fixed B⊥ = 0.5 T and D = 526 mV nm−1, forming a hysteresis loop as ν is swept past the phase boundary. d, Antisymmetrized ρxy measured as B⊥ is swept back and forth at fixed ν = 0.322, D = 526 mV nm−1. There is hysteresis when the sample enters/exits the TEC state, consistent with a first-order phase transition.
Extended Data Fig. 7 Doping and magnetic field dependence of the first-order phase transition.
a, b, Antisymmetrized ρxy(ν, D) normalized by the applied B⊥, and symmetrized ρxx(ν, D) at B⊥ = ±0.9 and 2.5 T, respectively. b is reproduced from Fig. 3a. Horizontal dashed lines mark D = 516, 500, and 485 mV nm−1. c, Landau fan of ρxy normalized by the applied B⊥ recorded by scanning ν from away (left panel, →) and towards (right panel, ←) charge neutrality point at each B⊥ and D = 516 mV nm−1. Dashed lines are a guide to the eye for the location of the phase boundary inferred from the data when ν is scanned away from the charge neutrality point. d, Landau fan of ρxx measured at the same time as c. e, f, and g, h, are the same as c,d but obtained at D = 500 and 485 mV nm−1, respectively. Both ρxx and ρxy exhibit hysteresis when varying ν induces a transition between the competing phases.
Extended Data Fig. 8 Gate–dependent resistance maps.
a, b, Antisymmetrized ρxy and symmetrized ρxx maps, respectively, obtained at B⊥ = 2.5 T and plotted against Vtg and Vbg. The horizontal dashed lines mark fixed Vbg = 4.95 V. The arrows show orthogonal axes of increasing n and D, related to Vtg and Vbg via the linear transformation described in the Methods. The dashed lines separate a region of normal metal with no broken isospin degeneracies at large negative Vbg from a region of full isospin degeneracy lifting (sometimes coexisting with translational symmetry breaking) at smaller values of Vbg.
Extended Data Fig. 9 Effect of an in-plane magnetic field on the states at ν = 1/4 and 1/3.
a, b, ρxx and ρxy maps, respectively, obtained at several values of B∥ with fixed B⊥ = 0.5 T, T = 10 mK. c, Maps of ρxx as a function of ν and B∥ measured at D = 527 mV nm−1 (denoted by the red dashed lines in a). An in-plane magnetic field stabilizes the \({C}_{1/4}^{-1}\) state, whereas the \({C}_{1/3}^{-1}\) state is suppressed. d, Measurements of − ρxy and ρxx versus B∥ at fixed ν = 0.319, obtained under the same conditions as c. Measurements are obtained at T = 10 mK, without (anti)symmetrization.
Extended Data Fig. 10 Time-series response of the Hall resistance with B⊥ and D sweeps.
a, ρxy (bottom) recorded at ν = 0.251 in response to a sequence of forward and reverse sweeps of B⊥ (top) and D (middle), with a fixed B∥ = 0.35 T. The sign of ρxy switches abruptly to align with the sign of B⊥, consistent with a C = +1 state. Excursions of D about 531 mV nm−1 can also switch the sign of ρxy (obtained with fixed B⊥ = −50 mT). b, c, Representative traces of ρxx and ρxy measured as D is swept back and forth with B⊥ = −50 mT. d-f, Analogous measurement to panels a-c, acquired with B∥ = 0. In this case, excursions of D about 531 mV nm−1 do not switch the sign of ρxy. g-i, Analogous measurements with B∥ = 2 T. In contrast to a-f, the sign of ρxy in response to B⊥ indicates a C = −1 state. Excursions of D about 531 mV nm−1 also do not change the sign of ρxy. All data was obtained at T = 300 mK.
Extended Data Fig. 11 Possible two-state model explaining the Chern number reversal of the TEC state at ν = 1/4.
Schematic diagrams showing a possible mechanism for the reversal in the sign of C across BPT, as analyzed in Fig. 2 of the main text, as well as its dependence on B∥ and D as analyzed in Fig. 4. The diagram on the left posits the existence of two closely competing TEC states at B⊥ = B∥ = 0, denoted by the blue and yellow markers. The blue circle represents the ground state since it has the lowest energy. The model is agnostic to the precise details of the two different states, but assumes that the magnetization of the state marked by the yellow square is larger than that of the state marked by the blue dot, and also that the direction of magnetization for a given sign of Chern number is opposite for the two states. The second assumption is plausible because there is, in general, no fixed relationship between the sign of C and the sign of the total magnetization for orbital magnetic states. The energies of the C = +1 and −1 branches of each state evolve oppositely in an applied out-of-plane field, and their magnetization is proportional to the slope of these lines. Above a certain value of B⊥, one branch of the state with larger magnetization (yellow square) has a lower energy than either branch of the original ground state (blue circle). The crossing points are denoted as “±PT”, indicating the value of B⊥ at which a first-order phase transition between the two states is anticipated. For each state, the sign of the Chern number of the branch with lower energy in an applied B⊥ corresponds to which has its total magnetization aligned with the external field. These are a minimal set of assumptions needed to explain the reversal in the sign of the Chern number across BPT. In this model, the additional reversal with B∥ could result from its coupling to the energetic hierarchy of the two states at fixed B⊥. For instance, B∥ could lower the energy of the state marked by the yellow square to a value lower than that of the state marked by the blue circle, thus making the former the ground state. A similar ground state reversal may also result from changing D, explaining the gate-induced hysteresis at fixed B⊥ and B∥. Although these models are consistent with all of our measurements, we cannot rule out alternative explanations not considered here.
Extended Data Fig. 12 Hartree–Fock calculations of the Chern number of the TEC state at ν = 1/4.
a, Calculated Chern number as a function of twist angle, θ, and potential difference, δ, with fixed tunneling parameter, α = 0.3, and dielectric constant, ϵ = 6 (see Methods for definitions of parameters). Parameters that do not yield a gap at ν = 1/4 are uncolored. At the twist angle appropriate for our sample, θ = 1.50°, the model predicts a C = 1 state for δ = 90 meV. We do not attempt to convert δ directly to the experimental electric displacement field, D, as this would require knowledge of the self-consistent electrostatics of the five graphene layers. We note that to a first approximation, the δ between layers depends on the applied displacement field D and the spacing between layers, δ ≈ 4eDdlayer, where 4dlayer is the spacing between the top of the bilayer and bottom of the trilayer. However, this conversion between δ and D is not to be taken to be precise: it will be modified by any charges that build up on different layers, which we know to be a significant effect. b, Calculated Chern number as a function of α and ϵ with fixed θ = 1.50° and δ = 90 meV.
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Su, R., Waters, D., Zhou, B. et al. Moiré-driven topological electronic crystals in twisted graphene. Nature 637, 1084–1089 (2025). https://doi.org/10.1038/s41586-024-08239-6
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DOI: https://doi.org/10.1038/s41586-024-08239-6
