====== $O(N)$ model ======

{{tag>3d holographic}}

The special cases of the [[o2_model]] and [[o3_model]] are particularly well-studied. Higher values of $N$ do come up occasionally, see for instance the [[o4_model]]. The [[Ising|Ising model]] is the case $N=1$, and the case $N=0$ is covered by  [[self_avoiding_walk|self-avoiding walks]].

This also goes by the name of the "n-vector model". Its Ginzburg-Landau description is the critical point of an O$(N)$-symmetric vector $\phi_i$ of scalar fields with a $\phi^4 =(\phi_i \phi_i)^2$ interaction.

Until 2025 this was thought to be the same CFT as the non-linear $\sigma$ model (NLSM), but recent research has challenged this identification: the NLSM has a protected operator of dimension $N-1$ which cannot be seen in the WF CFT [[https://arxiv.org/abs/2505.21611 | 1]] [[https://arxiv.org/abs/2602.10194 | 2]].

There are very similar versions of this CFT with different statistics.
The fermionic version is the [[https://arxiv.org/abs/1607.05316 | Gross-Neveu-Yukawa CFT]], which is the critical point of the U$(N)$-symmetric QFT of $N$ fermions and one scalar field.
There is also a very similar supersymmetric CFT with $N+1$ chiral superfields: see [[https://arxiv.org/pdf/1409.1937 | section 4.3]].

===== General dimension =====

This CFT is usually thought to exist between 2 and 4 dimensions. However, if we are willing to consider complex (i.e. nonunitary) CFTs, it also [[https://arxiv.org/abs/1910.02462 | exists in $d>4$]].

===== Large-N limit =====

[[https://arxiv.org/pdf/hep-th/0210114.pdf]]

===== External links =====

  * [[https://en.wikipedia.org/wiki/N-vector_model|Wikipedia]]