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Hofstadter’s butterfly, the predicted energy spectrum for non-interacting electrons confined to a two-dimensional lattice in a magnetic field, is one of the most remarkable fractal structures in nature1. At rational ratios of magnetic flux quanta per lattice unit cell, this spectrum shows self-similar distributions of energy levels that reflect its recursive construction. For most materials, Hofstadter’s butterfly is predicted under experimental conditions that are unachievable using laboratory-scale magnetic fields1,2,3. More recently, electrical transport studies have provided evidence for Hofstadter’s butterfly in materials engineered to have artificially large lattice constants4,5,6, such as those with moiré superlattices7,8,9,10. Yet, so far, direct spectroscopy of the fractal energy spectrum predicted by Hofstadter nearly 50 years ago has remained out of reach. Here we use high-resolution scanning tunnelling microscopy/spectroscopy (STM/STS) to investigate the flat electronic bands in twisted bilayer graphene (TBG) near the predicted second magic angle11,12, an ideal setting for spectroscopic studies of Hofstadter’s spectrum. Our study shows the fractionalization of flat moiré bands into discrete Hofstadter subbands and discerns experimental signatures of self-similarity of this spectrum. Moreover, our measurements uncover a spectrum that evolves dynamically with electron density, showing phenomena beyond that of Hofstadter’s original model owing to the combined effects of strong correlations, Coulomb interactions and the quantum degeneracy of electrons in TBG.

権利情報:

• In Copyright
  

  This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1038/s41586-024-08550-2.

キーワード: Hofstadter's butterfly, Twisted bilayer graphene (TBG), Scanning tunnelling microscopy/spectroscopy (STM/STS)

刊行年月日: 2025-03-06

出版者: Springer Science and Business Media LLC

掲載誌:

• Nature (ISSN: 00280836) vol. 639 issue. 8053 p. 60-66

研究助成金:

原稿種別: 著者最終稿 (Accepted manuscript)

MDR DOI:

公開URL: https://doi.org/10.1038/s41586-024-08550-2

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更新時刻: 2026-07-03 15:51:03 +0900

MDRでの公開時刻: 2026-07-03 18:30:20 +0900