# Spectroscopy of the fractal Hofstadter energy spectrum

https://mdr.nims.go.jp/datasets/fbc42664-3dc0-4045-a2f0-8d3f1a9eace0

## File

- [2025A00433G_Hofstadter_TBG_Main_241209.pdf](https://mdr.nims.go.jp/filesets/b97dea7c-392f-4b4e-9591-b6884d1adac8/download) ([Detail](https://mdr.nims.go.jp/filesets/b97dea7c-392f-4b4e-9591-b6884d1adac8.md))

## Id

fbc42664-3dc0-4045-a2f0-8d3f1a9eace0

## Local identifier



## Visibility

open_to_public

## State

published

## Created at

2026-07-02T01:45:03.306516Z

## Updated at

2026-07-03T06:51:03.639656Z

## Published at

2026-07-03T09:30:20.719862Z

## Doi



## First published url

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

## Date published

2025-03-06

## Recorded date published

2025-3-6

## Resource type

journal_article

## Manuscript type

accepted_manuscript

## Collection



## Title

- title: Spectroscopy of the fractal Hofstadter energy spectrum
  title_type: original
  lang: en

## Description

- description: 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.
  description_type: abstract
  lang: en

## Creator

- name: Kevin P. Nuckolls
  role: author
- name: Michael G. Scheer
  role: author
- name: Dillon Wong
  role: author
- name: Myungchul Oh
  role: author
- name: Ryan L. Lee
  role: author
- name: Jonah Herzog-Arbeitman
  role: author
- name: Kenji Watanabe
  role: author
  orcid: https://orcid.org/0000-0003-3701-8119
  organization: National Institute for Materials Science
- name: Takashi Taniguchi
  role: author
  orcid: https://orcid.org/0000-0002-1467-3105
  organization: National Institute for Materials Science
- name: Biao Lian
  role: author
- name: Ali Yazdani
  role: author

## Contact agent



## Publisher

organization: Springer Science and Business Media LLC

## Managing organization



## Keyword

- subject: Hofstadter's butterfly
  schema: not_defined
- subject: Twisted bilayer graphene (TBG)
  schema: not_defined
- subject: Scanning tunnelling microscopy/spectroscopy (STM/STS)
  schema: not_defined

## Rights

- description: '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.'
  identifier: http://rightsstatements.org/vocab/InC/1.0/

## Other identifier(s)



## Data origin

- data_origin_type: other

## Embargo

start_date: 2025-02-26
end_date: 2025-08-26

## Journal

- title: Nature
  issn: '00280836'
  volume: '639'
  issue: '8053'
  start_page: 60
  end_page: 66

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## Fileset

- id: b97dea7c-392f-4b4e-9591-b6884d1adac8
  filename: 2025A00433G_Hofstadter_TBG_Main_241209.pdf
  content_type: application/pdf
  size: 1899400
  md5: 9c6573f6204c86623f5729d900dd11df

## Thumbnail

fileset_id: b97dea7c-392f-4b4e-9591-b6884d1adac8
filename: 2025A00433G_Hofstadter_TBG_Main_241209.pdf