Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249 (1976).
Wannier, G. H. A result not dependent on rationality for Bloch electrons in a magnetic field. Phys. Status Solidi B 88, 757–765 (1978).
Streda, P. Quantised Hall effect in a two-dimensional periodic potential. J. Phys. C: Solid State Phys. 15, L1299 (1982).
Gerhardts, R. R., Weiss, D. & Wulf, U. Magnetoresistance oscillations in a grid potential: indication of a Hofstadter-type energy spectrum. Phys. Rev. B 43, 5192–5195 (1991).
Albrecht, C. et al. Evidence of Hofstadter’s fractal energy spectrum in the quantized Hall conductance. Phys. Rev. Lett. 86, 147–150 (2001).
Geisler, M. C. et al. Detection of a Landau band-coupling-induced rearrangement of the Hofstadter butterfly. Phys. Rev. Lett. 92, 256801 (2004).
Yankowitz, M. et al. Emergence of superlattice Dirac points in graphene on hexagonal boron nitride. Nat. Phys. 8, 382–386 (2012).
Dean, C. R. et al. Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices. Nature 497, 598–602 (2013).
Hunt, B. et al. Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure. Science 340, 1427–1430 (2013).
Ponomarenko, L. A. et al. Cloning of Dirac fermions in graphene superlattices. Nature 497, 594–597 (2013).
Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. 108, 12233–12237 (2011).
Lu, X. et al. Multiple flat bands and topological Hofstadter butterfly in twisted bilayer graphene close to the second magic angle. Proc. Natl Acad. Sci. 118, e2100006118 (2021).
Bistritzer, R. & MacDonald, A. H. Moiré butterflies in twisted bilayer graphene. Phys. Rev. B 84, 035440 (2011).
Yu, G. L. et al. Hierarchy of Hofstadter states and replica quantum Hall ferromagnetism in graphene superlattices. Nat. Phys. 10, 525–529 (2014).
Krishna Kumar, R. et al. High-temperature quantum oscillations caused by recurring Bloch states in graphene superlattices. Science 357, 181–184 (2017).
Barrier, J. et al. Long-range ballistic transport of Brown-Zak fermions in graphene superlattices. Nat. Commun. 11, 5756 (2020).
Yankowitz, M., Ma, Q., Jarillo-Herrero, P. & LeRoy, B. J. van der Waals heterostructures combining graphene and hexagonal boron nitride. Nat. Rev. Phys. 1, 112–125 (2019).
Spanton, E. M. et al. Observation of fractional Chern insulators in a van der Waals heterostructure. Science 360, 62–66 (2018).
Kometter, C. R. et al. Hofstadter states and re-entrant charge order in a semiconductor moiré lattice. Nat. Phys. 19, 1861–1867 (2023).
Yoo, H. et al. Atomic and electronic reconstruction at the van der Waals interface in twisted bilayer graphene. Nat. Mater. 18, 448–453 (2019).
Kazmierczak, N. P. et al. Strain fields in twisted bilayer graphene. Nat. Mater. 20, 956–963 (2021).
Wong, D. et al. Cascade of electronic transitions in magic-angle twisted bilayer graphene. Nature 582, 198–202 (2020).
Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).
Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).
Yankowitz, M. et al. Tuning superconductivity in twisted bilayer graphene. Science 363, 1059–1064 (2019).
Lu, X. et al. Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene. Nature 574, 653–657 (2019).
Chan, H. B., Glicofridis, P. I., Ashoori, R. C. & Melloch, M. R. Universal linear density of states for tunneling into the two-dimensional electron gas in a magnetic field. Phys. Rev. Lett. 79, 2867–2870 (1997).
Eisenstein, J. P., Pfeiffer, L. N. & West, K. W. Coulomb barrier to tunneling between parallel two-dimensional electron systems. Phys. Rev. Lett. 69, 3804–3807 (1992).
Liu, X. et al. Visualizing broken symmetry and topological defects in a quantum Hall ferromagnet. Science 375, 321–326 (2022).
Hu, Y. et al. High-resolution tunneling spectroscopy of fractional quantum Hall states. Preprint at https://doi.org/10.48550/arXiv.2308.05789 (2023).
MacDonald, A. H. Landau-level subband structure of electrons on a square lattice. Phys. Rev. B 28, 6713–6717 (1983).
Nuckolls, K. P. et al. Strongly correlated Chern insulators in magic-angle twisted bilayer graphene. Nature 588, 610–615 (2020).
Scheer, M. G., Herzog-Arbeitman, J., Nuckolls, K. P., Yazdani, A. & Lian, B. Hofstadter band theory for continuum models. (In preparation).
Bi, Z., Yuan, N. F. Q. & Fu, L. Designing flat bands by strain. Phys. Rev. B 100, 035448 (2019).
Lian, B., Xie, F. & Bernevig, B. A. Landau level of fragile topology. Phys. Rev. B 102, 041402 (2020).
Mandelbrot, B. How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 156, 636–638 (1967).
Falconer, K. Fractal Geometry: Mathematical Foundations and Applications (Wiley, 2014).
Dial, O. E., Ashoori, R. C., Pfeiffer, L. N. & West, K. W. High-resolution spectroscopy of two-dimensional electron systems. Nature 448, 176–179 (2007).
Nomura, K. & MacDonald, A. H. Quantum Hall ferromagnetism in graphene. Phys. Rev. Lett. 96, 256602 (2006).
Young, A. F. et al. Spin and valley quantum Hall ferromagnetism in graphene. Nat. Phys. 8, 550–556 (2012).
Herzog-Arbeitman, J., Song, Z.-D., Regnault, N. & Bernevig, B. A. Hofstadter topology: noncrystalline topological materials at high flux. Phys. Rev. Lett. 125, 236804 (2020).
Wong, D. et al. A modular ultra-high vacuum millikelvin scanning tunneling microscope. Rev. Sci. Instrum. 91, 023703 (2020).
