Measurement of the quantum metric of Landau levels in graphene

The dielectric response of insulators contains information about the quantum geometry of the underlying Hilbert space of electronic wavefunctions. In quantum Hall insulators, time-dependent polarization of Landau level wavefunctions leads to an intrinsic capacitive response due to the quantum metric - the real component of the quantum geometric tensor. Here, we probe this response by using microwave resonator spectroscopy to measure the real and imaginary parts of the AC conductivity of monolayer graphene quantum Hall insulators in a Corbino device geometry. We observe a step-like evolution of the dielectric constant at integer fillings, consistent with a quantum metric that scales as g~νl_B², where ν is the filling factor and l_B=√h/eB is the magnetic length. In partially filled Landau levels, we observe Wigner crystallization of dilute carriers stabilized by the quantum metric. Interpretations of the dielectric response in symmetry-broken-integer and fractional quantum Hall states are also discussed.