A Quantum Ruler for Topology and Quantum Geometry in Moiré Superlattices: Part 2/3

Flat and narrow band physics in moiré quantum matter (MQM) has proven to be extremely rich with new emergent quantum phases. The topological properties of the eigenstates of the moiré Hamiltonian are critical for establishing the quantum phase of the system. While the emergence of non-trivial Chern numbers has been observed, it is important to characterize the quantum geometry in detail including Berry curvature and less known quantum metric effects throughout the bands. Using a local probe, we employ magnetic oscillations as a “ruler” for quantum geometry in small-angle twisted double bilayer graphene (TDBG). Part 2: We perform gate-tuned scanning tunneling spectroscopy of the narrow moiré minibands in TDBG in magnetic fields up to B = 15 T to fully map the bands with varying displacement fields. The high-resolution Landau level spectra reveal tunable electron- and hole-like pockets that deviate significantly in their magnetic response from the semiclassical model. We demonstrate that these effects result from the quantum geometry in MQM and are specific to the typical MQM superlattice length scale.