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We study the anomalous Hall effect (AHE) for the double exchange model withthe exchange coupling $|J_H|$ being smaller than the bandwidth $|t|$ for thepurpose of clarifying the following unresolved and confusing issues: (i) theeffect of the underlying lattice structure, (ii) the relation between AHE andthe skyrmion number, (iii) the duality between real and momentum spaces, and(iv) the role of the disorder scatterings; which is more essential, $\sigma_H$(Hall conductivity) or $\rho_H$ (Hall resistivity)? Starting from a genericexpression for $\sigma_H$, we resolve all these issues and classify the regimesin the parameter space of $J_H \tau$ ($\tau$: elastic-scattering time), and$\lambda_{s}$ (length scale of spin texture). There are two distinct mechanismsof AHE; one is characterized by the real-space skyrmion-number, and the otherby momentum-space skyrmion-density at the Fermi level, which work in differentregimes of the parameter space.Comment: 4 pages, 1 figure, REVTe

Anomalous Hall Effect and Skyrmion Number in Real and Momentum Spaces