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COMPUTING SCIENCE

The World in a Spin

Brian Hayes

A Model Magnet

The Ising model was invented in 1920 by Wilhelm Lenz, who proposed it as a simplified version of a ferromagnet—the kind of magnet that holds your grocery list to the refrigerator door. A few years later Lenz's student Ernst Ising chose the model as the subject of his doctoral dissertation at the University of Hamburg.

The elements of the model are called spins, although the concept of rotation never enters the picture. You can think of a spin as nothing more than an arrow pointing either up or down (but in no other direction). The spins are arranged in a grid or lattice pattern. Spins at neighboring sites prefer to point the same way; in other words, the energy is lower when adjacent spins are parallel and higher when they are antiparallel. Except for these nearest-neighbor preferences, the spins don't interact at all. Thermal fluctuations tend to randomize the spins. Finally, an external magnetic field may impose a bias on the spin directions.

The Ising model is a crude cartoon of a ferromagnet, but it does capture the main features of the real thing. The Ising spins correspond to spinning electrons in iron atoms; the lattice represents the crystal structure; the nearest-neighbor interaction mimics the overlap of wave functions in adjacent iron atoms. The one element of the model that has no obvious counterpart in real physics is the requirement that spins take on only two possible orientations.

Is the model magnet magnetic? Do the spins line up in parallel the way they do in a real ferromagnet? It's easy to guess the answer at the extremes of the temperature range. At infinite temperature, thermal fluctuations overwhelm the nearest-neighbor interactions; each spin continually makes random flips, so that the average magnetization is zero. At the other end of the thermometer, thermal fluctuations disappear altogether at absolute zero, and the system falls into a state of minimum energy, with all the spins either up or down. (In the absence of an external field, the two choices are equally likely.)

Suppose we steadily reduce the temperature of an Ising model from infinity down to zero. Totally random spins must somehow become totally ordered, but is the change smooth and gradual, or does magnetization set in abruptly at some specific temperature? This question is an important one in statistical physics, where discontinuities—phase transitions and critical points—are major landmarks. For real ferromagnets, experiments give the answer. As iron cools from the melt, the magnetization remains zero until it suddenly leaps up at a temperature called the Curie point (about 1,040 kelvins). If the Ising model has anything to say about ferromagnetism, it should have a comparable discontinuity.

Ising's dissertation examined this question for a one-dimensional version of the model—a line of spins. His result was a disappointment. There was no phase transition at any temperature above zero. It's not hard to understand why. Consider a line of 10,000 spins, all pointing up. This configuration is clearly a minimum-energy state, since all nearest-neighbor pairs are parallel. Now reverse each of the 5,000 spins lying to the right of the midpoint. The overall magnetization falls to zero (since there are now equal numbers of up and down spins), and yet the change in the energy is minuscule: Only one pair of neighbors is pointing in opposite directions, while 9,998 pairs remain parallel. In a one-dimensional system, it seems, the coupling between spins is just too tenuous to overcome even the slightest thermal agitation.

Ising reportedly believed that his negative result would hold in higher dimensions as well. In this conjecture he was wrong. But before going on to recount the further history of the Ising model, I want to mention the further history of Ising. After receiving his doctorate, he taught physics in German public high schools, but as a Jew he was dismissed when Hitler came to power in 1933. He then taught at a Jewish boarding school in Potsdam, until that was destroyed in the Kristallnacht pogrom of 1938. Ising and his wife fled Germany, but they got only as far as Luxembourg before the war overtook them. There they managed to survive the occupation, and finally reached the United States in 1947. Ising taught physics and mathematics in Minot, North Dakota, and then for almost 30 years more at Bradley University in Peoria, Illinois. He died two years ago at age 98.








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