Spectroscopy of the fractal Hofstadter energy spectrum

Kevin P. Nuckolls, Michael G. Scheer, Dillon Wong, Myungchul Oh, Ryan L. Lee, Jonah Herzog-Arbeitman, Kenji Watanabe, Takashi Taniguchi, Biao Lian and Ali Yazdani ()
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Kevin P. Nuckolls: Princeton University
Michael G. Scheer: Princeton University
Dillon Wong: Princeton University
Myungchul Oh: Princeton University
Ryan L. Lee: Princeton University
Jonah Herzog-Arbeitman: Princeton University
Kenji Watanabe: National Institute for Materials Science
Takashi Taniguchi: National Institute for Materials Science
Biao Lian: Princeton University
Ali Yazdani: Princeton University

Nature, 2025, vol. 639, issue 8053, 60-66

Abstract: Abstract 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 fields1–3. More recently, electrical transport studies have provided evidence for Hofstadter’s butterfly in materials engineered to have artificially large lattice constants4–6, such as those with moiré superlattices7–10. 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.

Date: 2025
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DOI: 10.1038/s41586-024-08550-2

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