Topological Hall effect in weak coupling regime

Topological Hall effect (THE) arises from the exchange coupling M of conduction electrons to the magnetization texture having a scalar spin chirality. It is understood in terms of the effective magnetic field, B_eff, due to Berry phase in the strong coupling regime. In this case, electrons adjust their spin to the local magnetization, and B_eff is determined by the “local” magnetic structure. However, in the weak coupling regime, electrons fail in such adjustment and B_eff will be determined by a “nonlocal” spin chirality.
We investigate the THE in the weak coupling regime, M < γ (γ the electron scattering rate). In the diffusive regime (where the characteristic length scale, L, of the magnetic structure is longer than the electron mean free path), the locality of B_eff is determined by the relation among “spin precession length”, “spin diffusion length”, and L. When B_eff is “local”, the Hall conductivity is proportional to M, instead of the ordinary M³ dependence in the weakest coupling regime. For the “nonlocal” region, we applied the results to a skyrmion lattice and found that the Hall conductivity increases as the skyrmion size is increased; this behavior is opposite to the “local” case.