The fusion rules of the model are The fusion rules are invariant under the symmetry. The three-point structure constants are Knowing the fusion rules and three-point structure constants, it is possible to write operator product expansions, for example where are the conformal dimensions of the primary fields, and the omitted terms are contributions of descendant fields.
Correlation functions on the sphere
Any one-, two- and three-point function of primary fields is determined by conformal symmetry up to a multiplicative constant. This constant is set to be one for one- and two-point functions by a choice of field normalizations. The only non-trivial dynamical quantities are the three-point structure constants, which were given above in the context of operator product expansions. with. The three non-trivial four-point functions are of the type. For a four-point function, let and be the s- and t-channel Virasoro conformal blocks, which respectively correspond to the contributions of in the operator product expansion, and of in the operator product expansion. Let be the cross-ratio. In the case of, fusion rules allow only one primary field in all channels, namely the identity field. In the case of, fusion rules allow only the identity field in the s-channel, and the spin field in the t-channel. In the case of, fusion rules allow two primary fields in all channels: the identity field and the energy field. In this case we write the conformal blocks in the case only: the general case is obtained by inserting the prefactor, and identifying with the cross-ratio. In the case of, the conformal blocks are: From the representation of the model in terms of Dirac fermions, it is possible to compute correlation functions of any number of spin or energy operators: These formulas have generalizations to correlation functions on the torus, which involve theta functions.
Other observables
Disorder operator
The two-dimensional Ising model is mapped to itself by a high-low temperature duality. The image of the spin operator under this duality is a disorder operator, which has the same left and right conformal dimensions. Although the disorder operator does not belong to the minimal model, correlation functions involving the disorder operator can be computed exactly, for example whereas
Connectivities of clusters
The Ising model has a description as a random cluster model due to Fortuin and Kasteleyn. In this description, the natural observables are connectivities of clusters, i.e. probabilities that a number of points belong to the same cluster. The Ising model can then be viewed as the case of the -state Potts model, whose parameter can vary continuously, and is related to the central charge of the Virasoro algebra. In the critical limit, connectivities of clusters have the same behaviour under conformal transformations as correlation functions of the spin operator. Nevertheless, connectivities do not coincide with spin correlation functions: for example, the three-point connectivity does not vanish, while. There are four independent four-point connectivities, and their sum coincides with. Other combinations of four-point connectivities are not known analytically. In particular they are not related to correlation functions of the minimal model, although they are related to the limit of spin correlators in the -state Potts model.
