Condensed Matter Seminar

Hofstadter Hyperorbit Quantization of Landau Levels in twisted-Moiré Systems and Beyond

Dr. Dengyu Yang

National Institute for Standards and Technology (NIST)

Abstract: The Hofstadter butterfly is a striking quantum fractal: the energy spectrum of two-dimensional electrons moving through a periodic potential in a magnetic field. Its intricate structure connects quantum physics with number theory, with successive generations of spectral features organized by the Farey tree of rational numbers. Graphene moiré superlattices bring this physics within reach of laboratory magnetic fields, enabling electrical transport measurements to reveal topological gaps and their associated quantized Hall conductivities. However, transport does not directly resolve the energy bands that form the butterfly’s recursive structure. In this talk, I will present scanning tunneling spectroscopy measurements of a bilayer graphene–hexagonal boron nitride (hBN) moiré superlattice that resolve parent magnetic Bloch bands and their offspring, revealing the Farey-tree hierarchy in the electronic spectrum. I will also discuss our semiclassical analysis connecting the parent bands to their offspring through the quantization of Hofstadter hyperorbits. This analysis highlights the role of quantum geometry, through Berry curvature and the associated orbital magnetic moment, in determining hyperorbit energies. Together, these results connect the mathematical organization of the Hofstadter butterfly to the physical emergence of its nested spectral structure.