Anomalous Hall effect and skyrmion number in real and momentum spaces

We study the anomalous Hall effect (AHE) for the double exchange model with the exchange coupling \J(H)\ being smaller than the bandwidth \t\ for the purpose of clarifying the following unresolved and confusing issues: (i) the effect of the underlying lattice structure, (ii) the relation between AHE and the skyrmion number, (iii) the duality between real and momentum spaces, and (iv) the role of disorder scatterings; which is more essential, sigma(H) (Hall conductivity) or rho(H) (Hall resistivity)? Starting from a generic expression for sigma(H), we resolve all these issues and classify the regimes in the parameter space of J(H)tau (tau: elastic scattering time) and lambda(S) (length scale of spin texture). There are two distinct mechanisms of AHE; one is characterized by the real-space skyrmion number and the other by the momentum-space skyrmion density at the Fermi level, which occur in different regimes of the parameter space.