One-half topological number in entangled quantum physics

A topological phase can be engineered in quantum physics from the Bloch sphere of a spin-1/2 showing a hedgehog structure as a result of a radial magnetic field. We elaborate on a relation between the formation of an entangled wavefunction at one pole, in a two-spins model, and an interesting pair of one-half topological numbers. Similar to Cooper pairs in superconductors, an Einstein-Podolsky-Rosen pair or Bell state at one pole produces a half-flux quantum, which here refers to the halved flux of the Berry curvature on the surface. These 1/2 numbers also refer to the presence of a free Majorana fermion at a pole on each sphere. The topological responses can be measured when driving from north to south and from a circularly polarized field at the poles revealing the quantized or half-quantized nature of the protected transverse currents. We show applications of entangled wavefunctions in band structures, introducing a local marker in momentum space, to characterize the topological response of two-dimensional semimetals in bilayer geometries.