A quantum ruler for orbital magnetism in moiré quantum matter

For almost a century, magnetic oscillations have been a powerful “quantum ruler” for measuring Fermi surface topology. In this study, we used Landau-level spectroscopy to unravel the energy-resolved valley-contrasting orbital magnetism and large orbital magnetic susceptibility that contribute to the energies of Landau levels of twisted double-bilayer graphene. These orbital magnetism effects led to substantial deviations from the standard Onsager relation, which manifested as a breakdown in scaling of Landau-level orbits. These substantial magnetic responses emerged from the nontrivial quantum geometry of the electronic structure and the large length scale of the moiré lattice potential. Going beyond traditional measurements, Landau-level spectroscopy performed with a scanning tunneling microscope offers a complete quantum ruler that resolves the full energy dependence of orbital magnetic properties in moiré quantum matter. In a perpendicular magnetic field, electrons in two-dimensional (2D) materials occupy discrete energy levels known as Landau levels. These levels are described by the so-called Onsager relation. Slot et al . used scanning tunneling spectroscopy to measure the Landau levels in twisted double-bilayer graphene with an intermediate twist angle of 1.74° between the two bilayers. By comparing their results against the theoretical expectations, the researchers were able to measure deviations from the standard form of the Onsager relation stemming from the orbital magnetism in the system. —Jelena Stajic Scanning tunneling spectroscopy was used to measure the Landau levels in twisted double-bilayer graphene.