Surprises in the $O(N)$ models: nonperturbative fixed points, large $N$ limit and multi-criticality

We find that the multicritical fixed point structure of the O($N$) models is much more complicated than widely believed. In particular, we find new nonperturbative fixed points in three dimensions ($d=3$) as well as at $N=\mathrm{\infty }$. These fixed points come together with an intricate double-valued structure when they are considered as functions of $d$ and $N$. Many features found for the O($N$) models are shared by the O($N)\otimes$O(2) models relevant to frustrated magnetic systems.