O(N) model

An O(N) model has a real $N$-component order parameter $\mathbf{\varphi }$ with symmetry under the orthogonal group $O(N)$. A standard Landau-Ginzburg theory free energy isThe gradient term penalizes spatial variation, $r$ tunes the transition, and the positive quartic term bounds the local free energy below. The case $N=1$ has a sign symmetry; for $N>1$, rotations mix the components.

In $2<d<4$, the standard short-range model's continuous critical transition is governed by a Wilson-Fisher fixed point for the corresponding $N$-component order parameter. Near $d=4$, the epsilon expansion treats the interaction perturbatively because its engineering dimension is $4-d$. ^[ref]

The dimension restriction matters. At and above the upper critical dimension $d=4$, the Gaussian description controls the usual critical powers, with logarithmic qualifications at $d=4$. Lower-dimensional continuous-symmetry models can have different transition behavior, including the Berezinskii–Kosterlitz–Thouless transition. Thus the fixed-point claim must be matched to dimension, symmetry and the range of interactions.