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”, or “non-linear $\sigma$ model$.

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

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