  We propose a two-spin quantum-mechanical model with applied magnetic fields
acting on the Poincar\'e-Bloch sphere, to reveal a new class of topological
energy bands with Chern number one half for each spin-1/2. The mechanism behind
this fractional topology is a two-spin product state at the north pole and a
maximally entangled state close to the south pole. The fractional Chern number
of each spin can be measured through the magnetizations at the poles. We study
a precise protocol where the spin dynamics in time reflects the Landau-Zener
physics associated with energy band crossing effects. We show a correspondence
between the two-spin system and topological bilayer models on a honeycomb
lattice. These models describe semimetals with a nodal ring surrounding the
region of entanglement.
