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The Ising model, named after the physicist Ernst Ising, is a mathematical 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. The graph can exhibit periodic boundary conditions or free space boundary conditions depending on the system being modelled. 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 and the energy when they are aligned. It is also possible to include an external magnetic field. The Ising model is also used as a model of a simple liquid.

At a finite temperature, T, the probability of a configuration e with energy E(e) is proportional to

,

the whole thermodynamics being accessible from the partition function Z:

A full mathematical development of the Ising model and its solution in 1D is given in the article on the Potts model.

Visualization of the translation-invariant probability measure of the one-dimensional Ising model.

In his 1925 PhD thesis, Ising solved the model for the 1D case. In one dimension, the solution admits no phase transition. On the basis of this result, he incorrectly concluded that his model does not exhibit phase behaviour in any dimension.

Most numerical solutions involve using the Metropolis-Hastings algorithm running inside a Monte Carlo loop. Depending on the complexity only adjacent vertices can be taken into account or for long-range models other vertices can be included.

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.

While the Ising model is an extremely simplified description of ferromagnetism, its importance is underscored by the fact that other systems can be mapped exactly or approximately to the Ising system. The grand canonical ensemble formulation of the lattice gas model, for example, can be mapped exactly to the canonical ensemble formulation of the Ising model. The mapping allows one to exploit simulation and analytical results of the Ising model to answer questions about the related models.

The Ising model in two dimensions, and in the absence of an external magnetic field, was analytically solved at the critical point in 1944 by Lars Onsager but the 3D Ising model resisted solution for decades and was finally proved to be computationally intractable by Sorin Istrail in 2000 ^[1].

• Square-lattice Ising model
• Classical Heisenberg model
• Quantum Heisenberg model
• Kuramoto model
• XY model
• Potts model
• Maximal evenness
• Hopfield net
• ANNNI model
• Geometrical frustrated magnets
• t-J model

1. ^ Three-dimensional proof for Ising model impossible. Sandia National Laboratories (200-05-20). Retrieved on October 14, 2006.

• K. Binder, "Ising model" SpringerLink Encyclopaedia of Mathematics (2001)
• Barry M. McCoy and Tai Tsun Wu, The Two-Dimensional Ising Model, (1973) Harvard University Press, Cambridge Massachusetts, ISBN 0674914406
• Ross Kindermann and J. Laurie Snell, Random Markov Fields and Their Applications, (1980) American Mathematical Society, ISBN 0-8218-3381-2.
• "History of the Lenz-Ising Model" by Stephen G. Brush, Reviews of Modern Physics (American Physical Society) vol. 39, pp 883–893 (1967). (DOI: 10.1103/RevModPhys.39.883)

• 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.
• Science World article on the Ising Model
• An Ising Applet by Syracuse University
• A larger/more complicated 2D Ising Applet
• Phase transitions on lattices
• Nature news article: The Ising on the cake
• Three-dimensional proof for Ising Model impossible, Sandia researcher claims
• 3D Ising model simulation on GPU