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Hofstadter states and re-entrant charge order in a semiconductor moiré lattice


The emergence of moiré materials with flat bands provides a platform to systematically investigate and precisely control the correlated electronic phases. Here we report on a rich phase diagram of interpenetrating Hofstadter states—also called Chern insulators—and electron solids in a twisted WSe2/MoSe2 heterobilayer using local electronic compressibility measurements. We show that this reflects the presence of both flat and dispersive moiré bands whose relative energies, and therefore occupations, are tuned by the density and magnetic field. At low density, the competition between moiré bands leads to a transition from the commensurate arrangements of singlets at doubly occupied sites to triplet configurations at high fields. Hofstadter states are generally favoured at high density as dispersive bands are populated, but are suppressed by an intervening region of re-entrant charge-ordered states in which holes originating from multiple bands cooperatively crystallize. Our results reveal the key microscopic ingredients that favour distinct correlated ground states in semiconductor moiré systems, and they demonstrate an emergent lattice model system in which both interactions and band dispersion can be experimentally controlled.

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Fig. 1: Competing Hofstadter and charge-ordered states.
Fig. 2: Tuning between flat and dispersive moiré bands.
Fig. 3: Magnetic-field dependence of charge-ordered states.
Fig. 4: Interpenetrating Hofstadter states.

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Data availability

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

Code availability

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


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We thank T. Heinz, A. O’Beirne and H. B. Ribeiro for their assistance with the second-harmonic generation measurements, and T. P. Devereaux and A. A. Zibrov for helpful discussions. Experimental work was primarily supported by NSF-DMR-2103910. B.E.F. acknowledges an Alfred P. Sloan Foundation Fellowship and a Cottrell Scholar Award. The work at MIT was funded by the Air Force Office of Scientific Research (AFOSR) under award FA9550-22-1-0432. Y.Z. was supported by the start-up fund at the University of Tennessee. K.W. and T.T. acknowledge support from the JSPS KAKENHI (grant nos. 20H00354 and 23H02052) and World Premier International Research Center Initiative (WPI), MEXT, Japan. B.A.F. acknowledges a Stanford Graduate Fellowship. 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 and Affiliations



C.R.K. and J.Y. conducted the scanning SET measurements. C.R.K., J.Y. and B.E.F. designed the experiment. T.D., A.P.R., Y.Z. and L.F. conducted the theoretical calculations. C.R.K. fabricated the sample. K.W. and T.T. provided the hBN crystals. B.E.F. and L.F. supervised the project. All authors participated in the data analysis and writing of the manuscript.

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Correspondence to Benjamin E. Feldman.

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Nature Physics thanks Brian LeRoy and Eric Spanton for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Device image.

Optical microscope image of the MoSe2/WSe2 device. Platinum contacts to the sample, gold ‘contact’ gates to locally dope the heterobilayer, and the back gate electrode are all indicated. Scale bar: 10 μm.

Supplementary information

Supplementary Information

Supplementary Figs. 1–12 and discussion.

Source data

Source Data Fig. 2

Gap source data.

Source Data Fig. 3

Compressibility and gap source data.

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Kometter, C.R., Yu, J., Devakul, T. et al. Hofstadter states and re-entrant charge order in a semiconductor moiré lattice. Nat. Phys. 19, 1861–1867 (2023).

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