Spectroscopy of the Fractal Hofstadter Energy Spectrum
arXiv:2501.04777 · doi:10.1038/s41586-024-08550-2
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. Here we use high-resolution scanning tunneling microscopy / spectroscopy (STM / STS) to probe the flat electronic bands in twisted bilayer graphene near the predicted second magic angle, 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, 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.
20 pages, 5 figures
References in corpus (13)
- Emergence of Superlattice Dirac Points in Graphene on Hexagonal Boron Nitride
- Quantum Hall Ferromagnetism in Graphene
- Strongly Correlated Chern Insulators in Magic-Angle Twisted Bilayer Graphene
- Strain fields in twisted bilayer graphene
- Hierarchy of Hofstadter states and replica quantum Hall ferromagnetism in graphene superlattices
- High-temperature quantum oscillations caused by recurring Bloch states in graphene superlattices
- Visualizing Broken Symmetry and Topological Defects in a Quantum Hall Ferromagnet
- Hofstadter Topology: Non-crystalline Topological Materials at High Flux
- High Resolution Spectroscopy of Two-Dimensional Electron Systems
- Multiple Flat Bands and Topological Hofstadter Butterfly in Twisted Bilayer Graphene Close to the Second Magic Angle
- Long-range ballistic transport of Brown-Zak fermions in graphene superlattices
- A modular ultra-high vacuum millikelvin scanning tunneling microscope
- Hofstadter states and reentrant charge order in a semiconductor moiré lattice
Cited by in corpus (5)
- Microscopic Mechanism of Anyon Superconductivity Emerging from Fractional Chern Insulators
- Unification of Finite Symmetries in Simulation of Many-body Systems on Quantum Computers
- Quantization and quantum oscillations of the sublattice charge order in Dirac insulators
- Direct observation of flat bands in near-magic-angle twisted bilayer CVD graphene
- Mobility-edge-embedded Hofstadter butterfly from a tilt-induced quasiperiodic potential