Dielectric and optical markers originated from quantum geometry

We elaborate that practically all the non-excitonic dielectric and optical properties of semiconductors and insulators are determined by the quantum metric of the valence band states, including charge susceptibility, relative dielectric constant, optical conductivity, dielectric function, refractive index, absorption coefficient, reflectance, and transmittance. The key to this recognition is the complex optical conductivity, which contains the quantum metric in the optical transition matrix element, and the fact that all these dielectric and optical properties can be expressed in terms of the real and imaginary parts of optical conductivity. Our formalism allows to map all these properties to real space lattice sites as local markers following the formalism of topological markers, enabling the effect of disorder on the propagation of electromagnetic wave in the nanometer scale to be investigated, as demonstrated by a minimal model of 3D topological insulators.