Time-dependent Berry curvature and quantum metric of Floquet-Bloch states

Abstract

The quantum geometry of Bloch bands, characterized by the Berry curvature and the quantum metric, underpins a wide range of linear and nonlinear responses in static systems. Here, we extend this framework to periodically driven (Floquet) systems by introducing a time-dependent Berry curvature and quantum metric defined directly in the Floquet-Bloch basis. We derive optical sum rules that relate the Fourier components of these geometric quantities to the optical conductivity and demonstrate that, under ideal Floquet-band occupations, the first-order DC Hall and longitudinal responses at harmonic frequencies vanish identically. We further introduce a mixed Berry curvature involving time and momentum derivatives, which gives rise to a non-adiabatic quantized charge-pumping mechanism that occurs naturally during each driving period without requiring adiabatic evolution. In addition, we identify the time-domain quantum metric as a measure of the energy fluctuations of a Floquet band and interpret its mixed components as quantifying polarization-energy correlations. A comprehensive symmetry analysis reveals how time-reversal, sublattice (chiral), particle-hole, inversion, rotational, and reflection symmetries constrain the time-dependent quantum geometric tensor and its associated topological invariant. Numerical simulations of the Rudner-Lindner-Berg-Levin model and a fully symmetric Floquet model confirm the analytical predictions. These results establish the time-dependent quantum geometric tensor as a unified framework for describing the geometric, topological, and dynamical properties of periodically driven quantum systems, with direct implications for optical spectroscopy, quantum transport, and topological charge-pumping experiments.