Dirac Cones with Topological Quantized Response from Geometry, Light and Time

For topological states on the honeycomb lattice, we introduce a local formulation of the invariant $C^2$ from the Bloch sphere which can be revealed via the response of the Dirac cones to circularly polarized light. At resonance, we show its relation with the time evolution of the population associated to a topological energy band. For the quantum spin Hall effect, the $\mathbb{Z}_2$ topological number can be measured allowing a correspondence between the pfaffian and the robust response at the Dirac points. We address the topological protection towards a mass asymmetry and interactions. The photo-induced response can be measured in quantum materials, cold atomic and mesoscopic systems. The analogy between light and magnetic nuclear resonance may be applied for imaging.