The $N=3$ special case of the $O(N)$ model also goes by the name “Heisenberg model”.

There are two relevant operators, $\phi$ (an $O(3)$ vector) and $s$ (a singlet). The scaling dimensions are \[ \Delta_\phi = 0.51928(62) \text{ and } \Delta_s = 1.5957(55) \] while the nonvanishing OPE coefficients are \[ \lambda_{sss} = 0.499(12) \text{ and } \lambda_{\phi\phi s} = 0.5244(11) \text. \]

These data are from the bootstrap, computed here: https://arxiv.org/pdf/1603.04436.pdf.

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