A Digression on the Ising Model

Wilhelm Lenz (known for Lenz' law in electromagnetism), gave his graduate student Ernst Ising, the problem of determining the thermodynamic properties (e.g. the free energy and temperature of the phase transition) for a simple model of ferromagnetism. The energy expression for what became known as "the Ising model" is a special case of the equation

In the ferromagnetic Ising model, all the Jij are the same, a positive constant, and the summations are over nearest neighbor atoms only. In his doctoral thesis in 1924, Ising was only able to solve this problem for a one dimensional chain of atoms, which did not show any sign of a phase transition. He concluded that this simplified form of the interaction was not capable of producing a ferromagnetic phase transition.

He abandoned this line of research, and devoted himself to teaching. It wasn't until 1949 that he learned from the scientific literature that his paper had become widely known. He died at the age of 98 in 1998. Today, the Ising model is a widely used standard model of statistical physics. Every year, about 800 papers that apply this model are published, dealing with problems including neural networks, protein folding, biological membranes, social imitation, social impact in human societies and frustration.

There is a fairly simple proof in many statistical mechanics textbooks that shows that a one-dimensional Ising model can't have a phase transition. Other researchers suspected that there could be one in two or three dimensions, but the problem seemed intractable. The two-dimensional Ising model was finally solved in 1943 by one of the great names in statistical mechanics, Lars Onsager, who found an exact solution that had a phase transition. The solution of the antiferromagnetic Ising model (where the sign of the exchange constant is negative) can easily be obtained from the solution for the ferromagnet. There does not appear to be an exact solution in three dimensions, but the properties of the three-dimensional Ising model, and many variations including spin glasses, have now been thoroughly studied with computer simulations based on the Monte Carlo method.

Some interesting links: