Instantaneous response and quantum geometry of insulators

We present the time-dependent Quantum Geometric Tensor (tQGT) as a comprehensive tool for capturing the geometric character of insulators observable within linear response. We show that tQGT describes the zero-point motion of bound electrons and acts as a generating function for generalized sum rules of electronic conductivity. It therefore enables a systematic framework for computing the instantaneous response of insulators, including optical mass, orbital angular momentum, and dielectric constant. This construction guarantees a consistent approximation across these quantities upon restricting the number of occupied and unoccupied states in a low-energy description of an infinite quantum system. We outline how quantum geometry can be generated in periodic systems by lattice interference and examine spectral weight transfer from small frequencies to high frequencies by creating geometrically frustrated flat bands.