The two-dimensional critical Ising model is the critical limit of the Ising model in two dimensions.
It is a two-dimensional conformal field theory whose symmetry algebra is the Virasoro algebra with the central charge
Correlation functions of the spin and energy operators are described by the
While the minimal model has been exactly solved (see Ising critical exponents), the solution does not cover other observables such as connectivities of clusters.
minimal model is: This means that the space of states is generated by three primary states, which correspond to three primary fields or operators:[1] The decomposition of the space of states into irreducible representations of the product of the left- and right-moving Virasoro algebras is where
is the irreducible highest-weight representation of the Virasoro algebra with the conformal dimension
In particular, the Ising model is diagonal and unitary.
The characters of the three representations of the Virasoro algebra that appear in the space of states are[1] where
The modular invariant partition function is The fusion rules of the model are The fusion rules are invariant under the
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
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.
The three non-trivial four-point functions are of the type
be the s- and t-channel Virasoro conformal blocks, which respectively correspond to the contributions of
(and its descendants) in the operator product expansion
(and its descendants) in the operator product expansion
, fusion rules allow only the identity field in the s-channel, and the spin field in the t-channel.
only: the general case is obtained by inserting the prefactor
, 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:[1] These formulas have generalizations to correlation functions on the torus, which involve theta functions.
[1] The two-dimensional Ising model is mapped to itself by a high-low temperature duality.
Although the disorder operator does not belong to the minimal model, correlation functions involving the disorder operator can be computed exactly, for example[1] whereas 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
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
[3] Other combinations of four-point connectivities are not known analytically.