Spectroscopy of the Fractal Hofstadter Energy Spectrum

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 moir\'e 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 moir\'e 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.