Global and local topological quantized responses from geometry, light, and time

To describe a spin-$\frac{1}{2}$ particle on the Bloch sphere with a radial magnetic field and topological states of matter from the reciprocal space, we introduce $C$ square (${C}²$) as a local formulation of the global topological invariant. For the Haldane model on the honeycomb lattice, this ${C}²$ can be measured from the Dirac points through circularly polarized light related to the high-symmetry $M$ point(s). For the quantum spin Hall effect and the Kane-Mele model, the ${\mathbb{Z}}₂$ topological number robust to interactions can be measured locally from a correspondence between the Pfaffian and light. We address a relation with a spin pump and the quantum spin Hall conductance. The analogy between light and magnetic nuclear resonance may be applied for imaging, among other applications.