An introduction to the Ising model

This article is an invitation, or advertisement, for readers to work on a problem which is apparently very difficult, yet certainly extremely important. The problem is known generically as the Ising model, named after Ernst Ising, who did the first work on it in the early 1920s. Although unpromising in its initial results, the Ising model has turned out to be an exceptionally rich idea. The number of papers written on the subject is staggering; the number which remain to be written is conceivably even more staggering. The Ising model is concerned with the physics of phase transitions, which occur when a small change in a parameter such as temperature or pressure causes a large-scale, qualitative change in the state of a system. Phase transitions are common in physics and familiar in everyday life: we see one, for instance, whenever the temperature drops below 320F, and another whenever we put a kettle of water on the stove. Other examples include the formation of binary alloys and the phenomenon of ferromagnetism. The latter is also of interest historically: an understanding of ferromagnetism-and especially "spontaneous magnetization"was the original purpose of the Ising model and the subject of Ising's doctoral dissertation. Partly for this historical significance, we shall use ferromagnetism as a reference point later on for interpreting various features of the model. In spite of their familiarity, phase transitions are not well understood. One purpose of the Ising model is to explain how short-range interactions between, say, molecules in a crystal give rise to long-range, correlative behavior, and to predict in some sense the potential for a phase transition. The Ising model has also been applied to problems in chemistry, molecular biology, and other areas where "cooperative" behavior of large systems is studied. These applications are possible because the Ising model can be formulated as a mathematical problem. Although we shall refer frequently to the physics of ferromagnetism and use language from statistical mechanics, it is the mathematical aspects of the model which will concern us in this article. In particular we shall see that the Ising model has a combinatorial interpretation which is powerful enough in itself to establish some of the basic results concerning phase transitions. There are many other approaches and aspects to the