The simplest models, which become solvable in the limit of a large number of field components, deal with a field which has N components forming an O(N) vector in an internal symmetry space. A model of this kind was first considered by Stanley [Sta68] in statistical mechanics and is known as the spherical model. The extension to quantum field theory was made by Wilson [Wil73] both for the four-Fermi and ϕ4 theories.

Within the framework of perturbation theory, the four-Fermi interaction is renormalizable only in d = 2 dimensions and is nonrenormalizable for d > 2. The 1/N-expansion resums perturbation-theory diagrams after which the four-Fermi interaction becomes renormalizable to each order in 1/N for 2 ≤ d < 4. An analogous expansion exists for the nonlinear O(N) sigma model. The ϕ4 theory remains “trivial” in d = 4 to each order of the 1/N-expansion and has a nontrivial infrared-stable fixed point for 2 < d < 4.

The 1/N-expansion of the vector models is associated with a resummation of Feynman diagrams. A very simple class of diagrams – the bubble graphs – survives to the leading order in 1/N. This is why the large-N limit of the vector models is solvable. Alternatively, the large-N solution is nothing but a saddle-point solution in the path-integral approach. The existence of the saddle point is a result of the fact that N is large.