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.