Super-Instantons and the Reliability of Perturbation Theory in Non-Abelian Models

In dimension $D\leq 2$ the low temperature behavior of systems enjoying a continuous symmetry is dominated by super-instantons: classical configurations of arbitrarily low energy. Perturbation theory in the background of a super-instanton produces thermodynamic answers for the invariant Green's functions that differ from the standard ones, but only in non-Abelian models and only starting at $O(1/\beta^2)$. This effect modifies the $\beta$-function of the $O(N)$ models and persists in the large $N$ limit of the $O(N)$ models.