The Ising model is an interacting CFT in three dimensions. For related models in other dimensions, see other dimensions.

The Ising CFT also goes by the name “Wilson-Fisher fixed point” (in any $d < 4$). Another terminological difficulty: “Ising model” can mean either the lattice Ising model (discussed below) or the CFT itself. Even when the lattice model is tuned to criticality, these are different objects.

The lattice Ising model is usually defined on a three-dimensional rectangular lattice. A configuration labels site $i$ by a spin $s_i$. The action is given by \[ S = - J \sum_{\langle i j \rangle} s_i s_j \] and the partition function is of course the sum over all possible spin configurations $Z = \sum_s e^{-S(s)}$.

On this lattice, there is a critical J ($J_c \approx 2.7$) at which a second-order phase transition appears. This lattice model is not, of course, itself a CFT. Rather, once the coupling has been tuned to $J_c$, it flows to the Ising CFT in the IR. It is not unique in this respect; see for example scalar field theory.

An Ising model can of course be defined on any graph. There are many lattices with 3d translational symmetries (although discrete) but different local structure. Typically, an Ising model on these lattices will again have a critical point that flows to the Ising CFT in the IR.

One option is to keep the same graph, but change the geometry. This yields the anisotropic Ising model. \[ S = \sum_r \left[ J_x s_r s_{r+\hat x} + J_y s_r s_{r+\hat y} + J_z s_r s_{r+\hat z} \right] \] The case where $J_y = J_z \gg J_x$ is termed the Hamiltonian limit, as it is connected by the Suzuki-Trotter expansion (and the transfer matrix) to a quantum mechanical system. \[ H = -\mu \sum_i \sigma_x(i) -J \sum_{\langle i j \rangle} \sigma_z(i) \sigma_z(j) \] Note that the sum now goes over all sites on a two-dimensional “spatial” lattice. This model has a critical point at $\mu / J \sim 3$.

Scalar field theory, in the continuum, exhibits a second-order transition in the $\phi^4$ coupling. At this transition, the theory flows to the Ising CFT in the IR. This is true in both two and three dimensions. In 2d, see for instance 0902.0045 for a lattice measurement of the critical coupling.

On the lattice, there's a second (essentially unrelated) way to extract the Ising model from scalar field theory. Taking $m^2\rightarrow -\infty$ and $\lambda\rightarrow\infty$ simultaneously, two deep and narrow potential wells are created, and the fields are fixed to sit in one of these wells. In the limit, this recovers the lattice Ising model, with a coupling set by $\lambda/m^2$. Tuning that lattice Ising coupling yields the Ising CFT.

The liquid-gas transition in 3d is generally believed to belong to the Ising universality class. See for instance this determination of the critical exponents via molecular dynamics simulation.

As mentioned above, some properties of the liquid-gas transition are believed to be described by the Ising model.

Lattice Ising models can be defined in any number of dimensions. The critical point of such a theory is referred to as an Ising CFT in that number of dimensions. Only in three dimensions does an interacting theory result. The other possibilities are described below.

As usual, in one dimension the lattice Ising model is exactly solvable. This is most easily done through the transfer matrix. The corresponding quantum Hamiltonian is a two-level system.

The two-dimensional lattice Ising model was solved exactly by Onsager.

Critical points of lattice $\phi^4$ and Ising models are believed to be trivial in $d \ge 4$ dimensions.

In $d > 4$ dimensions this was proven by Michael Aizenman.

• An improved lattice measurement of the critical coupling in $\phi^4_2$ theory (Schaich)
• Geometric analysis of $\phi^4$ fields and Ising models (Aizenman)