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

Three of the leading operators in the spectrum have had their dimensions precisely determined: $\phi$ (an $O(3)$ vector), $s$ (a singlet), and $t$ (a symmetric rank-2 tensor). The scaling dimensions are \[ \Delta_\phi = 0.518936(\mathbf{67}),~\Delta_s = 1.59488(\mathbf{81}),~\Delta_t = 1.20954(\mathbf{32}). \]

Additionally, the rank-4 tensor $t_4$ has a rigorous upper bound for its scaling dimension: \[ \Delta_{t_4} < 2.99056. \] Notably, this result makes this operator relevant.

Five of OPE coefficients have also been determined, albeit not rigorously: \[ \lambda_{\phi\phi s} =0.524261(59),~\lambda_{s s s} = 0.5055(11),\\~\lambda_{t t s} = 0.98348(39),~\lambda_{\phi\phi t} = 0.87451(22),~\lambda_{t t t} = 1.49957(49). \]

The flavor current and stress tensor central charges have been determined nonrigorously: \[ C_J / C_J^{\text {free}}= 0.90632(16),~C_T / C_T^{\text {free}}= 0.944524(28). \]

These data are from the bootstrap, computed here: https://arxiv.org/pdf/2011.14647.pdf.

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