Interacting topological quantum aspects with light and geometrical functions

I review my recent progress and develop a geometrical approach in the quantum with light as a guide, from the vector potential in classical physics, revealing that topological properties can be equivalently measured from the poles of a sphere. The topological state is induced on the Bloch sphere of a spin-1/2 particle from a radial magnetic field related to the physics of Skyrmions. This shows a relation between the global topological response being measured at the poles, the response to a circularly polarized field and the quantum metric. I show how this approach is helpful for the classification of matter with the detection of the global topological invariant at specific points in the Brillouin zone, e.g. the Dirac points, from the responses to electromagnetic waves such as circularly polarized light and from new geometrical functions associated to the quantum metric measuring the quantum Hall and spin Hall conductivities. The $M$ point associated to the Brillouin zone of the honeycomb lattice also reveals the topological signature. Interactions are included in momentum space within a stochastic variational approach. In a realistic quantum model of interacting spins, this leads to fractional topological entangled aspects with a correspondence between a pair of half invariants and a Einstein–Podolsky–Rosen (EPR) pair or Bell state at one pole. I also formulate a correspondence between fractional topological numbers and resonating valence bond states. This approach gives further insight on the characterization of topological matter linked to superconductivity, protected topological semimetals in two dimensions and on the search of Majorana fermions for topologically protected quantum information. We also address a correspondence with the fractional quantum Hall effect and surface states of three-dimensional topological insulators.