  Nicolai maps offer an alternative description of supersymmetric theories via
nonlinear and nonlocal transformations characterized by the so-called
`free-action' and `determinant-matching' conditions. The latter expresses the
equality of the Jacobian determinant of the transformation with the one
obtained by integrating out the fermions, which so far have been considered
only to quadratic terms. We argue that such a restriction is not substantial,
as Nicolai maps can be constructed for arbitrary nonlinear sigma models, which
feature four-fermion interactions. The fermionic effective one-loop action then
gets generalized to higher loops and the perturbative tree expansion of such
Nicolai maps receives quantum corrections in the form of fermion loop
decorations. The `free-action condition' continues to hold for the classical
map, but the `determinant-matching condition' is extended to an infinite
hierarchy in fermion loop order. After general considerations for sigma models
in four dimensions, we specialize to the case of $\mathbb{C}\mathrm{P}^N$
symmetric spaces and construct the associated Nicolai map. These sigma models
admit a formulation with only quadratic fermions via an auxiliary vector field,
which does not simplify our analysis.
