Editing Conformal field theory (section)

=== 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"/>