It is interesting to view your question above not only in light of the Goldstone-Wilczek (G-W) approach (G-W has provided a method for computing the fermion charge induced by a classical profile), but also by computing 1/2-fermion charge found by [Jackiw-Rebbi](http://journals.aps.org/prd/abstract/10.1103/PhysRevD.13.3398) using G-W method. For simplicity, let us consider the 1+1D case, and let us consider the Z2 domain wall and the 1/2-charge found by Jackiw-Rebbi. The construction, valid for 1+1D systems, works as follows.
Consider a Lagrangian describing spinless fermions ψ(x) coupled to a classical background profile λ(x)
via a term λψ†σ3ψ. In the high temperature phase, the v.e.v. of λ is zero and no mass is generated
for the fermions. In the low temperature phase, the λ acquires two degenerate vacuum values ±⟨λ⟩
that are related by a Z2 symmetry. Generically we have
⟨λ⟩cos(ϕ(x)−θ(x)),
where we use the bosonization dictionary
ψ†σ3ψ→cos(ϕ(x))
and a phase change
Δθ=π captures the existence of a domain wall
separating regions with opposite values of the v.e.v. of
λ.
From the fact that the fermion density
ρ(x)=ψ†(x)ψ(x)=12π∂xϕ(x),
and the current
Jμ=ψ†γμψ=12πϵμν∂νϕ,
it follows that the induced charge
Qdw on the kink by a domain wall is
Qdw=∫x0+εx0−εdxρ(x)=∫x0+εx0−εdx12π∂xϕ(x)=12ππ=12,
where
x0 denotes the center of the domain wall.
You can try to extend to other dimensions, but then you may need to be careful and you may not be able to use the bosonization.
See more details of the derivation here in the [page 13](https://arxiv.org/abs/1403.5256) of [this paper](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.91.195134).