= O(N) model
{c}
An <O(N) model> has a real $N$-component <order parameter> $\boldsymbol\phi$ with symmetry under the <orthogonal group> $O(N)$. A standard <Landau-Ginzburg theory> <free energy> is
$$
F[\boldsymbol\phi]=\int d^dx\left[\frac\kappa2|\nabla\boldsymbol\phi|^2+\frac r2|\boldsymbol\phi|^2+\frac u4(|\boldsymbol\phi|^2)^2\right],\qquad\kappa,u>0.
$$
The 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 $4-d$. <Critical Exponents in 3.99 Dimensions by Kenneth Geddes Wilson and Michael Ellis Fisher>[1972 paper, small-epsilon interacting fixed-point construction; model normalization declared explicitly here]{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.
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