Spin textures with quantized winding numbers — topological Hall effect from emergent magnetic field
Skyrmions are topologically protected spin textures characterised by an integer winding number Q = (1/4π)∫n·(∂ₓn×∂ᵧn)dA. Each skyrmion contributes Q = −1.
The emergent magnetic field b ∝ n·(∂ₓn×∂ᵧn) deflects conduction electrons, producing the topological Hall effect even without spin-orbit coupling.
Dzyaloshinskii-Moriya interaction (D) from broken inversion symmetry stabilises the skyrmion lattice against the ferromagnetic exchange (J) and applied field (B).