The special cases of the $O(2)$ model and $O(3)$ model are particularly well-studied. Higher values of $N$ do come up occasionally, see for instance the $O(4)$ model. The Ising model is the case $N=1$, and the case $N=0$ is covered by self-avoiding walks.

This also goes by the name of the “n-vector model”. Its Ginzburg-Landau description is the critical point of an O$(N)$-symmetric vector $\phi_i$ of scalar fields with a $\phi^4 =(\phi_i \phi_i)^2$ interaction.

Until 2025 this was thought to be the same CFT as the non-linear $\sigma$ model (NLSM), but recent research has challenged this identification: the NLSM has a protected operator of dimension $N-1$ which cannot be seen in the WF CFT 1 2

https://arxiv.org/pdf/hep-th/0210114.pdf

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