The **Ising model**, named after Ernst Ising, is a model in statistical mechanics. It can be represented on a graph where its configuration space is the set of all possible assignments of +1 or -1 to each vertex of the graph. To complete the model, a function, *E(e)* must be defined, giving the difference between the energy of the "bond" associated with the edge when the spins on both ends of the bond are opposite than when they are aligned. It's also possible to have an external magnetic field.

At a finite temperature, *T*, the probability of a configuration is proportional to

- .

The Ising model undergoes a phase transition between an ordered and a disordered phase in 2 dimensions or more. In 2 dimensions, the Ising model has a strong/weak duality (between high temperatures and low ones) called the Kramers-Wannier duality. The fixed point of this duality is at the second-order phase transition temperature.

*See also:* XY model, Potts model

- Barry A. Cipra, "The Ising model is NP-complete", SIAM News, Vol. 33, No. 6; online edition (.pdf)
- Reports why the Ising model can't be solved exactly in general, since non-planar Ising models are NP-complete.

- Reports why the Ising model can't be solved exactly in general, since non-planar Ising models are NP-complete.