Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239\u20132249 (1976).<\/p>\n
Article<\/a>\u00a0 Shaffer, D., Wang, J. & Santos, L. H. Unconventional self-similar Hofstadter superconductivity from repulsive interactions. Nat. Commun. 13, 7785 (2022).<\/p>\n Article<\/a>\u00a0 Mishra, A., Hassan, S. R. & Shankar, R. Effects of interaction in the Hofstadter regime of the honeycomb lattice. Phys. Rev. B 93, 125134 (2016).<\/p>\n Article<\/a>\u00a0 Hong, S.-P. & Suck Salk, S.-H. Harper\u2019s equation for two-dimensional systems of antiferromagnetically correlated electrons. Phys. Rev. B 60, 9550\u20139554 (1999).<\/p>\n Article<\/a>\u00a0 M\u00f6ller, G. & Cooper, N. R. Fractional Chern insulators in Harper\u2013Hofstadter bands with higher Chern number. Phys. Rev. Lett. 115, 126401 (2015).<\/p>\n Article<\/a>\u00a0 Bistritzer, R. & MacDonald, A. H. Moir\u00e9 butterflies in twisted bilayer graphene. Phys. Rev. B 84, 035440 (2011).<\/p>\n Article<\/a>\u00a0 Moon, P. & Koshino, M. Energy spectrum and quantum Hall effect in twisted bilayer graphene. Phys. Rev. B 85, 195458 (2012).<\/p>\n Article<\/a>\u00a0 Dean, C. R. et al. Hofstadter\u2019s butterfly and the fractal quantum Hall effect in moir\u00e9 superlattices. Nature 497, 598\u2013602 (2013).<\/p>\n Article<\/a>\u00a0 Hunt, B. et al. Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure. Science 340, 1427\u20131430 (2013).<\/p>\n Article<\/a>\u00a0 Ponomarenko, L. A. et al. Cloning of Dirac fermions in graphene superlattices. Nature 497, 594\u2013597 (2013).<\/p>\n Article<\/a>\u00a0 Spanton, E. M. et al. Observation of fractional Chern insulators in a van der Waals heterostructure. Science 360, 62\u201366 (2018).<\/p>\n Article<\/a>\u00a0 Xu, X., Yao, W., Xiao, D. & Heinz, T. F. Spin and pseudospins in layered transition metal dichalcogenides. Nat. Phys. 10, 343\u2013350 (2014).<\/p>\n Article<\/a>\u00a0 Gustafsson, M. V. et al. Ambipolar Landau levels and strong band-selective carrier interactions in monolayer WSe2. Nat. Mater. 17, 411\u2013415 (2018).<\/p>\n Article<\/a>\u00a0 Shi, Q. et al. Odd- and even-denominator fractional quantum Hall states in monolayer WSe2. Nat. Nanotechnol. 15, 569\u2013573 (2020).<\/p>\n Article<\/a>\u00a0 Foutty, B. A. et al. Anomalous Landau level gaps near magnetic transitions in monolayer WSe2.Phys. Rev. X 14, 031018 (2024).<\/p>\n Kol\u00e1\u0159, K., Yang, K., von Oppen, F. & Mora, C. Hofstadter spectrum of Chern bands in twisted transition metal dichalcogenides. Phys. Rev. B 110, 115114 (2024).<\/p>\n Article<\/a>\u00a0 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).<\/p>\n Article<\/a>\u00a0 Zhao, C. et al. Hofstadter spectrum in a semiconductor moir\u00e9 lattice. Phys. Rev. B 111, L201302 (2025).<\/p>\n Article<\/a>\u00a0 Saito, Y. et al. Hofstadter subband ferromagnetism and symmetry-broken Chern insulators in twisted bilayer graphene. Nat. Phys. 17, 478\u2013481 (2021).<\/p>\n Article<\/a>\u00a0 Nuckolls, K. P. et al. Strongly correlated Chern insulators in magic-angle twisted bilayer graphene. Nature 588, 610\u2013615 (2020).<\/p>\n Article<\/a>\u00a0 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\u2013494 (2021).<\/p>\n Article<\/a>\u00a0 Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Flavour Hund\u2019s coupling, Chern gaps and charge diffusivity in moir\u00e9 graphene. Nature 592, 43\u201348 (2021).<\/p>\n Article<\/a>\u00a0 Das, I. et al. Symmetry-broken Chern insulators and Rashba-like Landau-level crossings in magic-angle bilayer graphene. Nat. Phys. 17, 710\u2013714 (2021).<\/p>\n Article<\/a>\u00a0 Yu, J. et al. Correlated Hofstadter spectrum and flavour phase diagram in magic-angle twisted bilayer graphene. Nat. Phys. 18, 825\u2013831 (2022).<\/p>\n Article<\/a>\u00a0 Mak, K. F. & Shan, J. Semiconductor moir\u00e9 materials. Nat. Nanotechnol. 17, 686\u2013695 (2022).<\/p>\n Article<\/a>\u00a0 Wang, L. et al. Correlated electronic phases in twisted bilayer transition metal dichalcogenides. Nat. Mater. 19, 861\u2013866 (2020).<\/p>\n Article<\/a>\u00a0 Ghiotto, A. et al. Quantum criticality in twisted transition metal dichalcogenides. Nature 597, 345\u2013349 (2021).<\/p>\n Article<\/a>\u00a0 Wu, F., Lovorn, T., Tutuc, E., Martin, I. & MacDonald, A. Topological insulators in twisted transition metal dichalcogenide homobilayers. Phys. Rev. Lett. 122, 086402 (2019).<\/p>\n Article<\/a>\u00a0 Devakul, T., Cr\u00e9pel, V., Zhang, Y. & Fu, L. Magic in twisted transition metal dichalcogenide bilayers. Nat. Commun. 12, 6730 (2021).<\/p>\n Article<\/a>\u00a0 Anderson, E. et al. Programming correlated magnetic states with gate-controlled moir\u00e9 geometry. Science 381, 325\u2013330 (2023).<\/p>\n Article<\/a>\u00a0 Foutty, B. A. et al. Mapping twist-tuned multiband topology in bilayer WSe2. Science 384, 343\u2013347 (2024).<\/p>\n Article<\/a>\u00a0 Cai, J. et al. Signatures of fractional quantum anomalous hall states in twisted MoTe2. Nature 622, 63\u201368 (2023).<\/p>\n Zeng, Y. et al. Thermodynamic evidence of fractional Chern insulator in moir\u00e9 MoTe2. Nature 622, 69\u201373 (2023).<\/p>\n Park, H. et al. Observation of fractionally quantized anomalous Hall effect. Nature 622, 74\u201379 (2023).<\/p>\n Xu, F. et al. Observation of integer and fractional quantum anomalous Hall effects in twisted bilayer MoTe2. Phys. Rev. X 13, 031037 (2023).<\/p>\n Guo, Y. et al. Superconductivity in 5.0o twisted bilayer WSe2. Nature 637, 839\u2013845 (2025).<\/p>\n Article<\/a>\u00a0 Xia, Y. et al. Superconductivity in twisted bilayer WSe2. Nature 637, 833\u2013838 (2025).<\/p>\n Article<\/a>\u00a0 Zhou, H. et al. Half- and quarter-metals in rhombohedral trilayer graphene. Nature 598, 429\u2013433 (2021).<\/p>\n Article<\/a>\u00a0 Zhao, W. et al. Realization of the Haldane Chern insulator in a moir\u00e9 lattice. Nat. Phys. 20, 275\u2013280 (2024).<\/p>\n Article<\/a>\u00a0 Kometter, C. R. et al. Hofstadter states and re-entrant charge order in a semiconductor moir\u00e9 lattice. Nat. Phys. 19, 1861\u20131867 (2023).<\/p>\n Ghiotto, A. et al. Stoner instabilities and Ising excitonic states in twisted transition metal dichalcogenides. Preprint at http:\/\/arxiv.org\/abs\/2405.17316<\/a> (2024).<\/p>\n Park, H. et al. Ferromagnetism and topology of the higher flat band in a fractional Chern insulator. Nat. Phys. 21, 549\u2013555 (2025).<\/p>\n Article<\/a>\u00a0 Kn\u00fcppel, P. et al. Correlated states controlled by a tunable van Hove singularity in moir\u00e9 WSe2 bilayers. Nat. Commun. 16, 1959 (2025).<\/p>\n Article<\/a>\u00a0 Streda, P. Theory of quantised Hall conductivity in two dimensions. J. Phys. C 15, L717 (1982).<\/p>\n Article<\/a>\u00a0 Eisenstein, J. P., Pfeiffer, L. N. & West, K. W. Negative compressibility of interacting two-dimensional electron and quasiparticle gases. Phys. Rev. Lett. 68, 674\u2013677 (1992).<\/p>\n Article<\/a>\u00a0 Feldman, B. E. et al. Fractional quantum hall phase transitions and four-flux states in graphene. Phys. Rev. Lett. 111, 076802 (2013).<\/p>\n Article<\/a>\u00a0 Zondiner, U. et al. Cascade of phase transitions and Dirac revivals in magic-angle graphene. Nature 582, 203\u2013208 (2020).<\/p>\n Article<\/a>\u00a0 Zhu, J., Su, J.-J. & MacDonald, A. Voltage-controlled magnetic reversal in orbital Chern insulators. Phys. Rev. Lett. 125, 227702 (2020).<\/p>\n Article<\/a>\u00a0 Redekop, E. et al. Direct magnetic imaging of fractional Chern insulators in twisted MoTe2. Nature 635, 584\u2013589 (2024).<\/p>\n Article<\/a>\u00a0 Zhang, X.-W. et al. Polarization-driven band topology evolution in twisted MoTe2 and WSe2. Nat. Commun. 15, 4223 (2024).<\/p>\n Article<\/a>\u00a0 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).<\/p>\n Foutty, B. A. et al. Tunable spin and valley excitations of correlated insulators in \u0393-valley moir\u00e9 bands. Nat. Mater. 22, 731\u2013736 (2023).<\/p>\n McGilly, L. J. et al. Visualization of moir\u00e9 superlattices. Nat. Nanotechnol. 15, 580\u2013584 (2020).<\/p>\n Article<\/a>\u00a0
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