The fascinating story of the Ising model

Great progress in our understanding of condensed matter physics has been made by the proposal and investigation of concrete models that are simple enough, to be amenable to mathematical analysis or to simulation on a computer, but complicated enough that they can capture the essential details of phenomena in a real material. This approach involves defining the energy of the whole system as a function of the different possible microscopic configurations of the system. Using a well-established theory called statistical mechanics it is possible, at least in principle, to connect the macroscopic properties of the system, such as how the heat capacity varies with temperature, to the possible microscopic configurations of the system.

The Ising model is a paradigm for this modeling approach. Every year thousands of journal articles are published that involve Ising models. They are applied to a wide range of topics in physics, chemistry, neuroscience, biology, sociology, and economics. In a magnetic system each magnetic atom is represented in the model by a single square box which can have two possible states, here represented as black or white, corresponding to two possible directions, such as ‘’up’’ or ``down’’ for the magnetic moment of the atom. These two directions can also be viewed as representing whether the magnetism of the atom is parallel or anti-parallel to the direction of an external magnetic field. In the model there is an energy gain when two adjoining boxes are in the same state, i.e. they are both black or both white. This energy gain captures a tendency towards ferromagnetism, i.e. where the direction of all the atomic magnets align with each other. In the absence of an external magnetic field, there is equal probability of a box being black or white. The random jiggling associated with the temperature of the system means the state of individual boxes are constantly changing. Figure 6.4. shows examples of likely states of the system for three different temperatures. Note that at very low temperatures, there is little jiggling, and the most likely state of the system is one where it is all black or all white. This is an example of the spontaneous symmetry breaking discussed in chapter 3, since in the model there is no preference for black or white.

The Ising model was originally proposed in 1920 by Wilhelm Lenz (1888-1957), a Professor at Hamburg University in Germany. He suggested investigation of the model for the doctoral research of a student, Ernst Ising (1900-1998).  In his thesis Ising was able to solve the one-dimensional version of the model exactly, i.e. calculate all the properties of the model without making any approximations in the mathematical analysis. He found that even for very low temperatures the model never undergoes a phase transition to an ordered ferromagnetic state. He also gave a rough argument that this would also be true in two and three dimensions. This led to doubts as to whether the model could describe the phase transition that occurs in ferromagnets such as iron, where at a critical temperature there is a transition to a state with no long-range magnetic order. However, Lars Onsager (1903-1976) performed a mathematical tour de force and in 1944 published an exact solution to the problem for a two-dimensional square lattice. The model did have a critical point at a non-zero temperature and Onsager calculated the critical exponents for the model. They were different from Landau’s theory (Chapter 3), raising questions about the validity of Landau’s theory.

The academic careers of the participants in this story are interesting. Lenz became an influential leader in theoretical physics in Germany. One of his other doctoral students, J. Hans D. Jensen (1907-1973) shared the Nobel Prize in Physics in 1963 for his work in theoretical nuclear physics. Although Ising is a household name in physics he did not go onto a distinguished academic career. After his doctorate, he worked in Germany as a high school teacher but lost his job because he was Jewish. He fled to Luxembourg and worked as a shepherd and a railway worker, before emigrating to the USA in 1947. He then worked until retirement teaching physics at a small university and did not resume research. He lived in Peoria, Illinois, which incidentally has become a metaphor for mainstream U.S.A., embodied in the question, ``Will it play in Peoria?"

Onsager was a Professor at Yale University and was awarded the Nobel Prize in Chemistry in 1968 for work on irreversible thermal processes. Yet, it is doubtful that Onsager would have survived in today’s ``publish or perish’’ academic culture. He only published one or two papers each year, and some were only a few pages long. He was also slow to publish. Often he would announce his new results at a conference, and then others would reference them in their own papers, and several years later Onsager would finally publish them. For example, he announced his solution of the Ising model two years before it was published. But many of his papers were seminal. Onsager was also known as being difficult to understand, even by brilliant colleagues graduate students. Before Onsager was hired by Yale, he was fired by both Brown University and John Hopkins University because his teaching was so poor.

Onsager’s solution of the Ising model provided a concrete example of an important idea: short-range interactions can lead to long-range order. Prior to Onsager’s solution, not everyone was convinced that this was true. In the Ising model each square (magnetic atom) only interacts directly with its nearest neighbours, i.e. the interactions are short-range. Yet in spite of  all the jiggling the whole system can form a state where changing the state of one atom is correlated with that of another atom infinitely far away, i.e. the system has long-range order. In different words, the system has rigidity, just like how pushing an atom on one side of a solid object will force even the atoms on the other side of the object to move.

Computer simulation of a two-dimensional Ising model as it passes through a critical point. The system shown here consists of a grid of 124 x 124 small boxes. Each box can be black or white. The probability of a box being black or white depends on the temperature and on whether its nearest neighbours are black or white. The left, centre, and right panels show a snapshot of a likely configuration of the system at a temperature less than, equal to, and greater than the critical temperature, Tc, respectively. Below the critical temperature (left panel) one sees the formation of large magnetic domains. There is more white than black representing spontaneous symmetry breaking. At the critical temperature (centre panel) there are equal numbers of black and white but there are paths through the whole system that pass through purely white or black, and so the range of correlations between squares becomes very large. Above the critical temperature, one cannot construct such paths, and the range of correlations is very short.

The figure is taken from this neuroscience paper.