Topological Matter and Fractional Entangled Quantum Geometry through Light

Here, we reveal our recent progress on a geometrical approach of quantum physics and topological crystals linking with Dirac magnetic monopoles and gauge fields through classical electrodynamics. The Bloch sphere of a quantum spin-1/2 particle acquires an integer topological charge in the presence of a radial magnetic field. We show that global topological properties are encoded from the poles of the surface allowing a correspondence between smooth fields, metric and quantum distance with the square of the topological number. The information is transported from each pole to the equatorial plane on a thin Dirac string. We develop the theory, "quantum topometry" in space and time, and present applications on transport from a Newtonian approach, on a quantized photo-electric effect from circular dichroism of light towards topological band structures of crystals. Edge modes related to topological lattice models are resolved analytically when deforming the sphere or ellipse onto a cylinder. Topological properties of the quantum Hall effect, quantum anomalous Hall effect and quantum spin Hall effect on the honeycomb lattice can be measured locally in the Brillouin zone from light-matter coupling. The formalism allows us to include interaction effects from the momentum space. Interactions may also result in fractional entangled geometry within the curved space. We develop a relation between entangled wavefunction in quantum mechanics, coherent superposition of geometries, a way to one-half topological numbers and Majorana fermions. We show realizations in topological matter. We present a link between axion electrodynamics, topological insulators on a surface of a cube and the two-spheres' model via merons.