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Topological bands and correlated states in helical trilayer graphene

Nature Physics volume 21, pages 239–244 (2025)Cite this article

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Abstract

The intrinsic anomalous Hall effect (AHE) is driven by non-zero Berry curvature and spontaneous time-reversal symmetry breaking. This effect can be realized in two-dimensional moiré systems hosting flat electronic bands but is not usually seen in inversion-symmetric materials. Here, we show that this physics is manifested in helical trilayer graphene—three graphene layers, each twisted in sequence by the same angle—although the system retains global in-plane inversion symmetry. We uncover a phase diagram of correlated and magnetic states at a magic twist angle of 1.8∘, which is explained by a lattice relaxation that leads to the formation of large periodic domains where in-plane inversion symmetry is broken on the moiré scale. Each domain harbours flat topological bands with opposite Chern numbers in the two valleys. We find correlated states at multiple integer and fractional electron fillings per moiré unit cell and an AHE at a subset of them. The AHE disappears above a critical electric displacement field at one electron per unit cell, indicating a topological phase transition. We establish helical trilayer graphene as a platform that presents an opportunity to engineer topology due to its emergent moiré-scale symmetries.

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Fig. 1: HTG.
Fig. 2: Strong electronic interactions.
Fig. 3: AHE.
Fig. 4: Density-induced switching near ν = 3.

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Data availability

All the experimental data used in this work are available via Harvard Dataverse at https://doi.org/10.7910/DVN/TVYXOI (ref. 67). Source data are provided with this paper. All other data that support the findings of this study are available from the corresponding authors upon reasonable request.

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Acknowledgements

We thank M. Kastner, P. Ledwith and E. Berg for helpful discussions, A. Bangura, G. Jones, R. Nowell, A. Woods and S. Hannahs for technical support and X. Wang for assistance with device fabrication. This work was partially supported by the Army Research Office MURI (grant no. W911NF2120147), the 2DMAGIC MURI (grant no. FA9550-19-1-0390), the National Science Foundation (grant no. DMR-1809802), the STC Center for Integrated Quantum Materials (NSF grant no. DMR-1231319), the Ramón Areces Foundation and the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant no. GBMF9463 to P.J.H. This work was supported by the Air Force Office of Scientific Research (AFOSR) under grant no. FA9550-22-1-0432. Measurement infrastructure was funded in part by the Gordon and Betty Moore Foundation’s EPiQS initiative through grant nos. GBMF3429 and GBMF9460. D.G.-G. gratefully acknowledges support from the Ross M. Brown Family Foundation. D.G.-G.’s involvement in measurements at Stanford and data analysis was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under contract DE-AC02-76SF00515. 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. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation cooperative agreement no. DMR-2128556 and the State of Florida. This work was performed in part at the Harvard University Center for Nanoscale Systems (CNS); a member of the National Nanotechnology Coordinated Infrastructure Network (NNCI), which is supported by the National Science Foundation under NSF grant no. ECCS-2025158. This work was carried out in part through the use of MIT.nano’s facilities. This work made use of the MRSEC Shared Experimental Facilities at MIT, supported by the National Science Foundation under grant no. DMR-1419807. A.U. acknowledges support from the MIT Pappalardo Fellowship and from the VATAT Outstanding Postdoctoral Fellowship in Quantum Science and Technology. Z.Z. is supported by a Stanford Science fellowship. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the US Department of Energy’s National Nuclear Security Administration under contract no. DE-NA000352.

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Author notes

  1. These authors contributed equally: Li-Qiao Xia, Sergio C. de la Barrera, Aviram Uri.

Authors and Affiliations

  1. Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Li-Qiao Xia, Sergio C. de la Barrera, Aviram Uri, Aaron Sharpe, Liang Fu, Trithep Devakul & Pablo Jarillo-Herrero

  2. Department of Physics, University of Toronto, Toronto, Ontario, Canada

    Sergio C. de la Barrera

  3. Materials Physics Department, Sandia National Laboratories, Livermore, CA, USA

    Aaron Sharpe

  4. Princeton Center for Theoretical Science, Princeton University, Princeton, NJ, USA

    Yves H. Kwan

  5. Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA, USA

    Ziyan Zhu & David Goldhaber-Gordon

  6. Research Center for Electronic and Optical Materials, National Institute for Materials Science, Tsukuba, Japan

    Kenji Watanabe

  7. Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Japan

    Takashi Taniguchi

  8. Department of Physics, Stanford University, Stanford, CA, USA

    David Goldhaber-Gordon & Trithep Devakul

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Contributions

S.C.d.l.B. and A.U. conceived the project. L.-Q.X. fabricated the devices with the help of A.U. L.-Q.X., A.U. and S.C.d.l.B. carried out the helium-3 transport measurements. A.S. and L.-Q.X. carried out the dilution refrigerator transport measurements under the supervision of D.G.-G. T.D. and Y.H.K. performed band structure, magnetization and Hartree–Fock calculations. T.D. and Z.Z. performed lattice-relaxation calculations. K.W. and T.T. supplied the boron nitride crystals. A.U., S.C.d.l.B., L.-Q.X., A.S., T.D., L.F. and P.J.-H. analysed the data and discussed the interpretation. A.U., S.C.d.l.B. and L.-Q.X. wrote the manuscript with input from all authors. P.J.-H. supervised the project.

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Correspondence to Li-Qiao Xia, Sergio C. de la Barrera, Aviram Uri or Pablo Jarillo-Herrero.

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Extended data

Extended Data Fig. 1 Optical micrographs of HTG devices.

a, Device 1 - a secondary device with θ = 1.77∘. b, Device 2 - our main device with θ = 1.79∘. c, Device 3. This device shares the van der Waals heterostructure with Device 2. Rxx and Ryx contacts are indicated by black dots for all devices. d, Contrast-enhanced optical micrograph of Device 2 after stacking. The crystallographic edges of the top hBN, bottom hBN, and top monolayer graphene are highlighted, showing no accidental alignment between hBN and HTG. All scale bars are 2 μm.

Extended Data Fig. 2 Twist angle determination.

a, Rxx Landau fan from Device 2, measured at D = 0 and T = 300 mK. Electron-side (right) and hole-side (left) are plotted with different color scales to improve contrast. Dashed lines correspond to the best-fit series shown in b. b, Map of the best-fit slopes from a emerging from a consistent set of integer fillings, ν. Red lines emerge from nν=±4 and ν = ± 4 (off-scale due to measurement limitations) in a and b, respectively.

Extended Data Fig. 3 Device 1 characterization.

a, Rxx versus n and D, showing resistance peaks at charge neutrality (ν = 0), at the moiré band gaps (ν = ± 4), and at the correlated states at ν=1,2,3. b, Field-trained ΔRyx measured at T = 300 mK and B = ± 60 mT versus ν and D. Hot spots near ν=1,3 indicate AHE. c,d, Field-antisymmetrized Ryx and field-symmetrized Rxx taken at ν=2.9 (cyan circle in b) and D/ϵ0 = − 0.15 V/nm while sweeping B up (solid) and down (dashed) at different temperatures as indicated. Temperature colorcode in d is identical to c. e,f, Same as c,d, taken at ν=0.8 and D/ϵ0 = 0.1 V/nm (pink triangle in b).

Extended Data Fig. 4 Device 3 characterization.

a, Rxx versus n and D, showing resistance peaks at charge neutrality (ν=0), at the moiré band gaps (ν = ± 4), and at the correlated states at ν=1,2,3. The contact resistance becomes very large when ν ≳ 3.2, leading to artifacts in the data. b, Field-trained ΔRyx measured at T = 300 mK and B = ± 60 mT versus ν and D. Hot spots near ν=1,3 indicate AHE. c,d, Field-antisymmetrized Ryx and field-symmetrized Rxx taken at ν=3.1 (cyan circle in b) and D/ϵ0 = 0 while sweeping B up (solid) and down (dashed) at different temperatures as indicated. The temperature color code in d is identical to c. e,f, Same as c,d, taken at ν=0.8 and D/ϵ0 = − 0.09 V/nm (pink triangle in b).

Extended Data Fig. 5 Single-particle density of states and Van Hove singularity.

(left) The single particle DOS with momentum-dependent tunneling as a function of filling factor and layer potential. The Van Hove singularity at which the Hall density switches sign is indicated by the dashed lines. (right) Extended Fermi surfaces at the Van Hove singularity are shown for the four points indicated by stars in the DOS plot.

Extended Data Fig. 6 Extraction of the Curie temperature using an Arrott plot.

\({R}_{yx}^{2}\) versus ∣B/Ryx∣. Positive (negative) extrapolated intercept of the linear part at high B indicates a ferromagnetic (paramagnetic) state. The curve taken at T = 10.5 K has approximately zero intercept, indicating a Curie temperature TC ≈ 10.5 K.

Extended Data Fig. 7 Temperature dependence and gap sizes estimation.

a, Rxx (raw data, not field-symmetrized) versus ν and T of Device 2 at D = 0 and B = 0. The jumps in resistance near ν=3 reflect the AHE of different magnetic states combined with Ryx mixing. A pronounced electron-hole asymmetry is demonstrated. b, line cuts of a versus 1/T at different ν as indicated. Some curves show non-monotonic temperature dependence due to the parallel conduction of the insulating domains and the metallic network of domain walls. The total resistance can be modelled by \({R}_{{{\rm{tot}}}}={({R}_{{{\rm{M}}}}^{-1}+{R}_{{{\rm{I}}}}^{-1})}^{-1}\), where RM and RI are the resistance of the metallic walls and insulating domains, respectively. When one of the resistances is much smaller than the other, we have \({R}_{{{\rm{tot}}}}\approx \min ({R}_{{{\rm{M}}}},{R}_{{{\rm{I}}}})\). At low temperatures, 1/T ≳ 0.2 K−1, the insulating bulk is shunted by the metal, saturating the increased resistance of the insulating domains. As T is increased to intermediate values, 1/T ~ 0.1 K−1, the domains become less resistive and Rtot ≈ RI. We use this regime to estimate the gap sizes for ν=0,2,3 (indicated) by fitting the data to \(R={R}_{0}\exp \{-{\Delta }_{\nu }/2{k}_{B}T\}\), where Δν is the gap at filling ν and kB is Boltzmann’s constant. At higher temperatures, 1/T ≲ 0.05 K−1 for ν=2,3, the correlated states give way to a metallic phase, accounting for the increasing resistance with T at those filling factors. At ν=1, evidence for thermal activation is absent, indicating a semimetallic state without a fully developed correlated gap.

Extended Data Fig. 8 Correlated state at ν=7/2.

a, Rxx versus ν, measured on Device 2 at D = 0, B = 0, and T = 300 mK. At ν=7/2, we find a resistance peak distinct from the one at ν=3. b, Waterfall plot of antisymmetrized Ryx taken by sweeping B up (solid) and down (dashed) as the fast axis at D = 0 and different ν, as indicated on the right of every other curve. The AHE persists beyond ν=7/2.

Extended Data Fig. 9 Hartree-Fock calculations at ν=7/2 and 2/3.

a, Charge density n(r) (measured relative to that at full flat band filling nν=+4(r)) of the tetrahedral antiferromagnet (TAF) at ν=7/2. Grey dots indicate ABA-stacking regions. b, Local spin orientation in the TAF. Arrows denote spin direction in sx − sy plane, while red (blue) coloring indicates out-of-plane polarization along \(+{\hat{s}}_{z}(-{\hat{s}}_{z})\). Grey parallelogram indicates the new quadrupled moiré unit cell. c,d Same as a except for the \({\hat{C}}_{3z}\) charge density wave and stripe charge density wave respectively. e, ΔE of the different translation symmetry breaking solutions at ν=7/2 as a function of interlayer potential U. ΔE is measured relative to that of the best translation- symmetric solution. f, Charge gap of the translation symmetry breaking solutions at ν=7/2. g,h, Same as e,f except for the best translation symmetry breaking solution at ν=2/3. All calculations performed on a 18 × 18 system using θ = 1.80∘, wAA = 75 meV.

Supplementary information

Supplementary Information

Supplementary Figs.1–10, discussion and Tables 1 and 2.

Source data

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Xia, LQ., de la Barrera, S.C., Uri, A. et al. Topological bands and correlated states in helical trilayer graphene. Nat. Phys. 21, 239–244 (2025). https://doi.org/10.1038/s41567-024-02731-6

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