  We prove that a normalized sequence of multiple Wigner integrals (in a fixed
order of free Wigner chaos) converges in law to the standard semicircular
distribution if and only if the corresponding sequence of fourth moments
converges to 2, the fourth moment of the semicircular law. This extends to the
free probabilistic, setting some recent results by Nualart and Peccati on
characterizations of central limit theorems in a fixed order of Gaussian Wiener
chaos. Our proof is combinatorial, analyzing the relevant noncrossing
partitions that control the moments of the integrals. We can also use these
techniques to distinguish the first order of chaos from all others in terms of
distributions; we then use tools from the free Malliavin calculus to give
quantitative bounds on a distance between different orders of chaos. When
applied to highly symmetric kernels, our results yield a new transfer
principle, connecting central limit theorems in free Wigner chaos to those in
Gaussian Wiener chaos. We use this to prove a new free version of an important
classical theorem, the Breuer-Major theorem.
