Any good book on statistical field theory and critical phenomena will have a chapter on large N. See for instance Zinn-Justin's book.
Keep in mind that the large N approximation is a special kind of mean-field theory (since it is both self-consistent and exact in large N). It is thus quite different from the usual mean-field theory use for example in the case of the Ising model.
It is called mean-field because one only have to minimize the action without computing any corrections (which are of order 1/N). But the equation is self-consistent, contrary to the usual mean-field theory.
To summarize : for the O(N) model, with N→∞, one could use the usual mean-field, or the "large N mean-field". The results would not be the same, as the former is approximate but the latter is exact.
In principle, you can also use the large N results to finite N, even though it is not exact in that case (but sometimes 3≫1). But it definitely does not work for the Ising model (N=1). Indeed, the large N approach use the fact that the Goldstone modes (the πi) dominate the physics compare to the fluctuation of σ (because there is an infinite number of Goldstone modes compare to one σ). But in the Ising case, the only mode is always gapped (away from criticality), so the results of the large N is non-sense.
This post imported from StackExchange Physics at 2014-03-05 14:32 (UCT), posted by SE-user Adam