Quantum twisting microscopy as a momentum-resolved probe of quantum geometry and sublattice symmetries

In two-dimensional (2D) flat-band systems, as engineered, for instance, by stacking and twisting van der Waals materials, the momentum dependence of the Bloch states is a key factor in determining the interaction-induced emergence of different phases. Hence, methods of probing this form of ''quantum geometry'' are crucial to understanding these systems. Yet, extracting the momentum dependence of the underlying quantum geometric tensor from measurements is challenging in 2D. Here, we propose employing a quantum twisting microscope (QTM) with a sublattice-polarized tip to fill this gap. We show how expectation values of sublattice Pauli matrices can be probed across the scanned momentum path. This allows for the direct measurement of crucial symmetries characterizing the different candidate instabilities in graphene-based moiré systems. We furthermore show that the quantum metric component along the scan direction can also be extracted; the full quantum geometric tensor is also accessible, given that more than one sublattice polarization can be used. Our work illustrates the potential and extends the versatility of the QTM approach. If implemented in future experiments, this would provide unprecedented access to the symmetry signatures of the correlated phases and to the underlying quantum geometry in twisted bilayer graphene and related systems.