Geometric contribution to the longitudinal conductivity in insulators

In recent years, quantum geometry of the Hilbert space became a unified language to describe various optical and condensed matter phenomena. However, compared to the better-known Berry curvature, we hear less about the quantum (Fubini-Study) metric g_αβ as it is hard to observe directly. g_αβ is nevertheless of the utmost importance as it parameterizes the quantum distance between states in the Hilbert space and can be related to the spread of the wavefunctions, which is of interest for unconventional superconductivity and Quantum Hall physics.

Geometry plays an especially important role when only two bands contribute to the conductivity σ_μν. Here, we capitalize on that, showing that in a Landau-level system, the imaginary transverse AC conductivity couples directly to the quantum metric, which suggests it is possible to measure g_αβ using current Hall devices. In the more general case where multiple bands participate in transport, the response is no longer proportional to the quantum metric, however, different bounds on Im(σ_μμ) in terms of the quantum geometry can be imposed.