  Topological Hall effect (THE) caused by a noncoplanar spin texture
characterized by a scalar spin chirality is often described by the Berry phase,
or the associated effective magnetic field. This picture is appropriate when
the coupling, $M$, of conduction electrons to the spin texture is strong
(strong-coupling regime) and the adiabatic condition is satisfied. However, in
the weak-coupling regime, where the coupling $M$ is smaller than the electrons'
scattering rate, the adiabatic condition is not satisfied and the Berry phase
picture does not hold. In such a regime, the relation of the effective magnetic
field to the spin texture can be "nonlocal", in contrast to the "local"
relation in the strong coupling case. Focusing on the case of continuous but
general spin texture, we investigate the THE in various characteristic regions
in the weak-coupling regime, namely, (1) diffusive and local, (2) diffusive and
nonlocal, and (3) ballistic. In the presence of spin relaxation, there arise
two more regions in the "weakest-coupling" regime: (1') diffusive and local,
and (2') diffusive and nonlocal. We derived the analytic expression of the
topological Hall conductivity (THC) for each region, and found that the
condition for the locality of the effective field is governed by transverse
spin diffusion of electrons. In region 1, where the spin relaxation is
negligible and the effective field is local, the THC is found to be
proportional to $M$, instead of $M^3$ of the weakest-coupling regime. In the
diffusive, nonlocal regions (2 and 2'), the effective field is given by a spin
chirality formed by "effective spins" that the electrons see during their
diffusive motion. Applying the results to a skyrmion lattice, we found the THC
decreses as the skyrmion density is increased in region 2', reflecting the
nonlocality of the effective field, and shows a maximum at the boundary to the
"local" region.
