We consider a special case of the $n$-component cubic model on the square lattice, for which an expansion exists in Ising-type graphs. We construct a transfer matrix and perform a finite-size-scaling analysis to determine the critical points for several values of $n$. Furthermore we determine several universal quantities, including three critical exponents. For $n<2$, these results agree well with the theoretical predictions for the critical $\mathrm{O}\left(n\right)$ branch. This model is also a special case of the $(N_{\alpha },N_{\beta })$ model of Domany and Riedel. It appears that the self-dual plane of the latter model contains the exactly known critical points of the $n=1$ and 2 cubic models. For this reason we have checked whether this is also the case for $1<n<2$. However, this possibility is excluded by our numerical results.

• Received 12 October 2005

DOI:https://doi.org/10.1103/PhysRevE.73.026104