Nonadiabatic quantum geometry and optical conductivity

The ground-state quantum geometry is at the root of several static and adiabatic properties, while genuinely dynamic properties are routinely addressed via Kubo formulas, whose essential entries are the excited states. It is shown here that the ground-state metric-curvature tensor evolves in time by means of a causal unitary operator, which by construction elucidates the geometrical effect of the excited states in compact form. In the condensed-matter case the generalized tensor encompasses the whole conductivity tensor at arbitrary frequencies in both insulators and metals, with the exception of the Drude term in the metallic case; the latter is shown to be eminently nongeometrical.