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

To describe a spin-12 particle on the Bloch sphere with a radial magnetic field and topological states of matter from the reciprocal space, we introduce C square (C2) as a local formulation of the global topological invariant. For the Haldane model on the honeycomb lattice, this C2 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 Z2 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.