Editing Conformal field theory (section)

== Examples ==
=== Mean field theory ===
A '''generalized free field''' is a field whose correlation functions are deduced from its two-point function by [[Wick's theorem]]. For instance, if <math>\phi</math> is a scalar primary field of dimension {{tmath|1= \Delta }}, its four-point function reads<ref name="fkps12"/>
: <math>
\left\langle \prod_{i=1}^4\phi(x_i) \right\rangle = \frac{1}{|x_{12}|^{2\Delta}|x_{34}|^{2\Delta}} + \frac{1}{|x_{13}|^{2\Delta}|x_{24}|^{2\Delta}} + \frac{1}{|x_{14}|^{2\Delta}|x_{23}|^{2\Delta}}.
</math>
For instance, if <math>\phi_1,\phi_2</math> are two scalar primary fields such that <math>\langle \phi_1\phi_2\rangle=0</math> (which is the case in particular if <math>\Delta_1\neq\Delta_2</math>), we have the four-point function
: <math>
\Big\langle \phi_1(x_1)\phi_1(x_2)\phi_2(x_3)\phi_2(x_4)\Big\rangle = \frac{1}{|x_{12}|^{2\Delta_1}|x_{34}|^{2\Delta_2}}.
</math>
'''Mean field theory''' is a generic name for conformal field theories that are built from generalized free fields. For example, a mean field theory can be built from one scalar primary field {{tmath|1= \phi }}. Then this theory contains {{tmath|1= \phi }}, its descendant fields, and the fields that appear in the OPE {{math|1= \phi \phi }}. The primary fields that appear in <math>\phi \phi</math> can be determined by decomposing the four-point function <math>\langle\phi\phi\phi\phi\rangle</math> in conformal blocks:<ref name="fkps12"/> their conformal dimensions belong to <math>2\Delta+2\mathbb{N}</math>: in mean field theory,  the conformal dimension is conserved modulo integers. Structure constants can be computed exactly in terms of the [[Gamma function]].<ref name="kks18"/>

Similarly, it is possible to construct mean field theories starting from a field with non-trivial Lorentz spin. For example, the 4d [[Maxwell theory]] (in the absence of charged matter fields) is a mean field theory built out of an antisymmetric tensor field <math>F_{\mu \nu}</math> with scaling dimension {{tmath|1= \Delta = 2 }}.

Mean field theories have a Lagrangian description in terms of a quadratic action involving Laplacian raised to an arbitrary real power (which determines the scaling dimension of the field). For a generic scaling dimension, the power of the Laplacian is non-integer. The corresponding mean field theory is then non-local (e.g. it does not have a conserved stress tensor operator).{{Citation needed|date=May 2021}}

=== Critical Ising model ===

The '''critical Ising model''' is the critical point of the [[Ising model]] on a hypercubic lattice in two or three dimensions. It has a <math>\mathbb{Z}_2</math> global symmetry, corresponding to flipping all spins. The [[two-dimensional critical Ising model]] includes the <math>\mathcal{M}(4,3)</math> [[Virasoro minimal model]], which can be solved exactly. There is no Ising CFT in <math>d \geq 4</math> dimensions.

=== Critical Potts model ===

The '''critical Potts model''' with <math>q=2,3,4,\cdots</math> colors is a unitary CFT that is invariant under the [[permutation group]] {{tmath|1= S_q }}. It is a generalization of the critical Ising model, which corresponds to {{tmath|1= q=2 }}. The critical Potts model exists in a range of dimensions depending on {{tmath|1= q }}.

The critical Potts model may be constructed as the [[continuum limit]] of the [[Potts model]] on ''d''-dimensional hypercubic lattice. In the Fortuin-Kasteleyn reformulation in terms of clusters, the Potts model can be defined for {{tmath|1= q\in\mathbb{C} }}, but it is not unitary if <math>q</math> is not integer.

=== Critical O(''N'') model ===

The '''critical O(''N'') model''' is a CFT invariant under the [[orthogonal group]]. For any integer {{tmath|1= N }}, it exists as an interacting, unitary and compact CFT in <math>d=3</math> dimensions (and for <math>N=1</math> also in two dimensions). It is a generalization of the critical Ising model, which corresponds to the O(N) CFT at {{tmath|1= N=1 }}.

The O(''N'') CFT can be constructed as the [[continuum limit]] of a lattice model with spins that are ''N''-vectors, called the [[n-vector model]].

Alternatively, the critical <math>O(N)</math> model can be constructed as the <math>\varepsilon \to 1</math> limit of [[Wilson–Fisher fixed point]] in <math>d=4-\varepsilon</math> dimensions. At {{tmath|1= \varepsilon = 0 }}, the Wilson–Fisher fixed point becomes the tensor product of <math>N</math> free scalars with dimension {{tmath|1= \Delta = 1 }}. For <math>0 < \varepsilon < 1</math> the model in question is non-unitary.<ref>{{cite journal|last1=Hogervorst|first1=Matthijs|last2=Rychkov|first2=Slava|last3=van Rees|first3=Balt C.|date=2016-06-20|title=Unitarity violation at the Wilson-Fisher fixed point in 4 − ε dimensions|journal=Physical Review D|language=en|volume=93|issue=12|pages=125025|arxiv=1512.00013|doi=10.1103/PhysRevD.93.125025|bibcode=2016PhRvD..93l5025H |s2cid=55817425|issn=2470-0010}}</ref>

When ''N'' is large, the O(''N'') model can be solved perturbatively in a 1/''N'' expansion by means of the [[Hubbard–Stratonovich transformation]]. In particular, the <math>N \to \infty</math> limit of the critical O(''N'') model is well-understood.

The conformal data of the critical O(''N'') model are functions of ''N'' and of the dimension, on which many results are known.<ref name="hen22"/>

=== Conformal gauge theories ===

Some conformal field theories in three and four dimensions admit a Lagrangian description in the form of a [[gauge theory]], either abelian or non-abelian. Examples of such CFTs are '''conformal QED''' with sufficiently many charged fields in <math>d=3</math> or the [[Banks-Zaks fixed point]] in {{tmath|1= d=4 }}.