The quantum geometric origin of capacitance in insulators*

Recent years have seen a significant focus on topological insulators famous for their unique electronic Hilbert space properties, often referred to as quantum geometry. Among these properties, the quantum metric is highly notable for its relevance in multiband superconductors among numerous other applications. Consequently, there is a growing demand to comprehend the features of the quantum metric and develop methods for its measurement.

We establish that this quantity can be routinely accessed through quasi-adiabatic AC transport probes. Specifically, we suggest that the quantum metric quantifies the oscillator strength of the electric dipole moment induced by the AC electric field, and serves as a main ingredient in determining the magnitude of the dielectric constant. This insight allows us to predict: (i) the quantization of the capacitance in Landau levels, (ii) an enhanced bulk capacitance in simple Chern insulators, and (iii) the sensitivity of the capacitance to the twist angle in twisted bilayer graphene.