We study the critical behavior of the $m$-anisotropic $n$-vector model with $n$ and $m$ both as continuous variables [$0\le n,m<\mathrm{}4-O\left(\epsilon \right)$, $\epsilon =4-d$, $d$ = dimensionality of space] to first order in $\epsilon$. The limit $n\to 0$, $m>\mathrm{}0\mathrm{}$ of the model is of interest as a model of self-avoiding rings and of polymerization. For $n\ge m,n,m$ integers, the critical behavior of the model is known to be that of the isotropic $m$-vector model, i.e., the $O\mathrm{}\left(m\right)\mathrm{}$ model. Here we prove that the critical behavior of the anisotropic model is always identical with that of the $O\mathrm{}\left(m\right)\mathrm{}$ model for real $n$,$m$ regardless of whether $n\ge m or n<m$. In particular, we prove that a single self-avoiding ring and a single self-avoiding walk belong to the same universality class of the $O\mathrm{}\left(0\right)\mathrm{}$ model, while polymerization belongs to the universality class of the $O\mathrm{}\left(m\right)\mathrm{}$ model, $m>\mathrm{}0\mathrm{}$.

• Received 28 September 1982

DOI:https://doi.org/10.1103/PhysRevB.27.4507