Greene Award: Visualizing the Fractal Hofstadter Energy Spectrum

ORAL  · Invited

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 nature. 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 fields. More recently, electrical transport studies have provided evidence for Hofstadter's butterfly in materials engineered to have artificially large lattice constants, such as those with moiré superlattices. Yet to-date, direct spectroscopy of the fractal energy spectrum predicted by Hofstadter nearly 50 years ago has remained out of reach. In this talk, I will discuss experiments using scanning tunneling microscopy / spectroscopy (STM / STS) that probe Hofstadter's spectrum directly [1]. Our study shows the fractionalization of flat moiré bands in twisted bilayer graphene 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, displaying phenomena beyond that of Hofstadter's original model due to the combined effects of strong correlations, Coulomb interactions, and the quantum degeneracy of electrons in twisted bilayer graphene.

[1] K. P. Nuckolls et al. arXiv preprint arXiv:2501.04777 (2025). (to appear in Nature)

*This work was primarily supported by the DOE-BES grant DE-FG02-07ER46419 to A.Y. Support for the theory-experimental collaboration was provided by the NSF-MRSEC grant NSFDMR-2011750 to A.Y. and B.L. Additional support was provided by the Gordon and Betty Moore Foundation (GBMF) grant GBMF9469 and the U.S. Army Research Office MURI project under grant number W911NF-21-2-0147, ONR grant N000142412471, and NSF grant DMR-2312311 to A.Y. M. S. and B. L. are supported by the National Science Foundation under award DMR-2141966. Additional support is provided by the GBMF grant GBMF8685 towards the Princeton theory program.

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Publication: K. P. Nuckolls et al. arXiv preprint arXiv:2501.04777 (2025).

Presenters

  • Kevin P Nuckolls

    • Princeton University
    • Massachusetts Institute of Technology

Authors

  • Kevin P Nuckolls

    • Princeton University
    • Massachusetts Institute of Technology

  • Michael G Scheer

    • Princeton University

  • Dillon Wong

    • Princeton University

  • Myungchul Oh

    • Pohang Univ of Sci & Tech

  • Ryan Li-Chung Lee

    • Princeton University

  • Jonah Herzog-Arbeitman

    • Princeton University

  • Kenji Watanabe

    • National Institute for Materials Science
    • NIMS
    • Research Center for Functional Materials, National Institute for Materials Science
    • Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan
    • Research Center for Functional Materials, National Institute of Material Science, Tsukuba, Japan
    • National Institute of Materials Science
    • Advanced Materials Laboratory, National Institute for Materials Science

  • Takashi Taniguchi

    • National Institute for Materials Science
    • International Center for Materials Nanoarchitectonics, National Institute for Materials Science
    • Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan
    • International Center for Materials Nanoarchitectonics, National Institute of Material Science, Tsukuba, Japan
    • Advanced Materials Laboratory, National Institute for Materials Science

  • Biao Lian

    • Princeton University

  • Ali Yazdani

    • Princeton University