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Self-similarity and fractals have fascinated researchers across variousdisciplines. In graphene placed on boron nitride and subjected to a magneticfield, self-similarity appears in the form of numerous replicas of the originalDirac spectrum, and their quantization gives rise to a fractal pattern ofLandau levels, referred to as the Hofstadter butterfly. Here we employcapacitance spectroscopy to probe directly the density of states (DoS) andenergy gaps in this spectrum. Without a magnetic field, replica spectra areseen as pronounced DoS minima surrounded by van Hove singularities. TheHofstadter butterfly shows up as recurring Landau fan diagrams in high fields.Electron-electron interactions add another twist to the self-similar behaviour.We observe suppression of quantum Hall ferromagnetism, a reverse Stonertransition at commensurable fluxes and additional ferromagnetism within replicaspectra. The strength and variety of the interaction effects indicate a largeplayground to study many-body physics in fractal Dirac systems.

Hierarchy of Hofstadter states and replica quantum Hall ferromagnetism in graphene superlattices