Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249 (1976).

Article 
ADS 

Google Scholar
 

Shaffer, D., Wang, J. & Santos, L. H. Unconventional self-similar Hofstadter superconductivity from repulsive interactions. Nat. Commun. 13, 7785 (2022).

Article 
ADS 

Google Scholar
 

Mishra, A., Hassan, S. R. & Shankar, R. Effects of interaction in the Hofstadter regime of the honeycomb lattice. Phys. Rev. B 93, 125134 (2016).

Article 
ADS 

Google Scholar
 

Hong, S.-P. & Suck Salk, S.-H. Harper’s equation for two-dimensional systems of antiferromagnetically correlated electrons. Phys. Rev. B 60, 9550–9554 (1999).

Article 
ADS 

Google Scholar
 

Möller, G. & Cooper, N. R. Fractional Chern insulators in Harper–Hofstadter bands with higher Chern number. Phys. Rev. Lett. 115, 126401 (2015).

Article 
ADS 

Google Scholar
 

Bistritzer, R. & MacDonald, A. H. Moiré butterflies in twisted bilayer graphene. Phys. Rev. B 84, 035440 (2011).

Article 
ADS 

Google Scholar
 

Moon, P. & Koshino, M. Energy spectrum and quantum Hall effect in twisted bilayer graphene. Phys. Rev. B 85, 195458 (2012).

Article 
ADS 

Google Scholar
 

Dean, C. R. et al. Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices. Nature 497, 598–602 (2013).

Article 
ADS 

Google Scholar
 

Hunt, B. et al. Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure. Science 340, 1427–1430 (2013).

Article 
ADS 

Google Scholar
 

Ponomarenko, L. A. et al. Cloning of Dirac fermions in graphene superlattices. Nature 497, 594–597 (2013).

Article 
ADS 

Google Scholar
 

Spanton, E. M. et al. Observation of fractional Chern insulators in a van der Waals heterostructure. Science 360, 62–66 (2018).

Article 
ADS 

Google Scholar
 

Xu, X., Yao, W., Xiao, D. & Heinz, T. F. Spin and pseudospins in layered transition metal dichalcogenides. Nat. Phys. 10, 343–350 (2014).

Article 

Google Scholar
 

Gustafsson, M. V. et al. Ambipolar Landau levels and strong band-selective carrier interactions in monolayer WSe2. Nat. Mater. 17, 411–415 (2018).

Article 
ADS 

Google Scholar
 

Shi, Q. et al. Odd- and even-denominator fractional quantum Hall states in monolayer WSe2. Nat. Nanotechnol. 15, 569–573 (2020).

Article 
ADS 

Google Scholar
 

Foutty, B. A. et al. Anomalous Landau level gaps near magnetic transitions in monolayer WSe2.Phys. Rev. X 14, 031018 (2024).


Google Scholar
 

Kolář, K., Yang, K., von Oppen, F. & Mora, C. Hofstadter spectrum of Chern bands in twisted transition metal dichalcogenides. Phys. Rev. B 110, 115114 (2024).

Article 
ADS 

Google Scholar
 

Wang, M., Wang, X. & Vafek, O. Phase diagram of twisted bilayer MoTe2 in a magnetic field with an account for the electron-electron interaction.Phys. Rev. B 110, L201107 (2024).

Article 
ADS 

Google Scholar
 

Zhao, C. et al. Hofstadter spectrum in a semiconductor moiré lattice. Phys. Rev. B 111, L201302 (2025).

Article 
ADS 

Google Scholar
 

Saito, Y. et al. Hofstadter subband ferromagnetism and symmetry-broken Chern insulators in twisted bilayer graphene. Nat. Phys. 17, 478–481 (2021).

Article 

Google Scholar
 

Nuckolls, K. P. et al. Strongly correlated Chern insulators in magic-angle twisted bilayer graphene. Nature 588, 610–615 (2020).

Article 
ADS 

Google Scholar
 

Wu, S., Zhang, Z., Watanabe, K., Taniguchi, T. & Andrei, E. Y. Chern insulators, van Hove singularities and topological flat bands in magic-angle twisted bilayer graphene. Nat. Mater. 20, 488–494 (2021).

Article 
ADS 

Google Scholar
 

Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Flavour Hund’s coupling, Chern gaps and charge diffusivity in moiré graphene. Nature 592, 43–48 (2021).

Article 

Google Scholar
 

Das, I. et al. Symmetry-broken Chern insulators and Rashba-like Landau-level crossings in magic-angle bilayer graphene. Nat. Phys. 17, 710–714 (2021).

Article 

Google Scholar
 

Yu, J. et al. Correlated Hofstadter spectrum and flavour phase diagram in magic-angle twisted bilayer graphene. Nat. Phys. 18, 825–831 (2022).

Article 

Google Scholar
 

Mak, K. F. & Shan, J. Semiconductor moiré materials. Nat. Nanotechnol. 17, 686–695 (2022).

Article 
ADS 

Google Scholar
 

Wang, L. et al. Correlated electronic phases in twisted bilayer transition metal dichalcogenides. Nat. Mater. 19, 861–866 (2020).

Article 
ADS 

Google Scholar
 

Ghiotto, A. et al. Quantum criticality in twisted transition metal dichalcogenides. Nature 597, 345–349 (2021).

Article 
ADS 

Google Scholar
 

Wu, F., Lovorn, T., Tutuc, E., Martin, I. & MacDonald, A. Topological insulators in twisted transition metal dichalcogenide homobilayers. Phys. Rev. Lett. 122, 086402 (2019).

Article 
ADS 

Google Scholar
 

Devakul, T., Crépel, V., Zhang, Y. & Fu, L. Magic in twisted transition metal dichalcogenide bilayers. Nat. Commun. 12, 6730 (2021).

Article 
ADS 

Google Scholar
 

Anderson, E. et al. Programming correlated magnetic states with gate-controlled moiré geometry. Science 381, 325–330 (2023).

Article 
ADS 

Google Scholar
 

Foutty, B. A. et al. Mapping twist-tuned multiband topology in bilayer WSe2. Science 384, 343–347 (2024).

Article 
ADS 

Google Scholar
 

Cai, J. et al. Signatures of fractional quantum anomalous hall states in twisted MoTe2. Nature 622, 63–68 (2023).

Zeng, Y. et al. Thermodynamic evidence of fractional Chern insulator in moiré MoTe2. Nature 622, 69–73 (2023).

Park, H. et al. Observation of fractionally quantized anomalous Hall effect. Nature 622, 74–79 (2023).

Xu, F. et al. Observation of integer and fractional quantum anomalous Hall effects in twisted bilayer MoTe2. Phys. Rev. X 13, 031037 (2023).


Google Scholar
 

Guo, Y. et al. Superconductivity in 5.0o twisted bilayer WSe2. Nature 637, 839–845 (2025).

Article 
ADS 

Google Scholar
 

Xia, Y. et al. Superconductivity in twisted bilayer WSe2. Nature 637, 833–838 (2025).

Article 
ADS 

Google Scholar
 

Zhou, H. et al. Half- and quarter-metals in rhombohedral trilayer graphene. Nature 598, 429–433 (2021).

Article 
ADS 

Google Scholar
 

Zhao, W. et al. Realization of the Haldane Chern insulator in a moiré lattice. Nat. Phys. 20, 275–280 (2024).

Article 

Google Scholar
 

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

Ghiotto, A. et al. Stoner instabilities and Ising excitonic states in twisted transition metal dichalcogenides. Preprint at http://arxiv.org/abs/2405.17316 (2024).

Park, H. et al. Ferromagnetism and topology of the higher flat band in a fractional Chern insulator. Nat. Phys. 21, 549–555 (2025).

Article 

Google Scholar
 

Knüppel, P. et al. Correlated states controlled by a tunable van Hove singularity in moiré WSe2 bilayers. Nat. Commun. 16, 1959 (2025).

Article 
ADS 

Google Scholar
 

Streda, P. Theory of quantised Hall conductivity in two dimensions. J. Phys. C 15, L717 (1982).

Article 
ADS 

Google Scholar
 

Eisenstein, J. P., Pfeiffer, L. N. & West, K. W. Negative compressibility of interacting two-dimensional electron and quasiparticle gases. Phys. Rev. Lett. 68, 674–677 (1992).

Article 
ADS 

Google Scholar
 

Feldman, B. E. et al. Fractional quantum hall phase transitions and four-flux states in graphene. Phys. Rev. Lett. 111, 076802 (2013).

Article 
ADS 

Google Scholar
 

Zondiner, U. et al. Cascade of phase transitions and Dirac revivals in magic-angle graphene. Nature 582, 203–208 (2020).

Article 
ADS 

Google Scholar
 

Zhu, J., Su, J.-J. & MacDonald, A. Voltage-controlled magnetic reversal in orbital Chern insulators. Phys. Rev. Lett. 125, 227702 (2020).

Article 
ADS 

Google Scholar
 

Redekop, E. et al. Direct magnetic imaging of fractional Chern insulators in twisted MoTe2. Nature 635, 584–589 (2024).

Article 
ADS 

Google Scholar
 

Zhang, X.-W. et al. Polarization-driven band topology evolution in twisted MoTe2 and WSe2. Nat. Commun. 15, 4223 (2024).

Article 
ADS 

Google Scholar
 

Qiu, W.-X., Li, B., Luo, X.-J. & Wu, F. Interaction-driven topological phase diagram of twisted bilayer MoTe2. Phys. Rev. X 13, 041026 (2023).


Google Scholar
 

Foutty, B. A. et al. Tunable spin and valley excitations of correlated insulators in Γ-valley moiré bands. Nat. Mater. 22, 731–736 (2023).

McGilly, L. J. et al. Visualization of moiré superlattices. Nat. Nanotechnol. 15, 580–584 (2020).

Article 
ADS 

Google Scholar