Electrons dance to a quantum beat in the Hubbard model of solid-state physics
At the most superficial level, the antics of electrons in solids seem easy enough to comprehend. A substance is conductive if at least some of its electrons are free to wander about; when every electron is tightly bound to an atom, the material is an insulator. Magnetic effects derive from the spin of the electron, which gives rise to a tiny magnetic dipole moment, like that of a bar magnet. The strongest form of magnetism (ferromagnetism) appears when nearly all the spins line up in the same direction.
Going beyond this level of understanding is not so easy. What property of a solid determines whether electrons are nomadic or frozen in place? Why do electron spins in iron and nickel adopt a parallel alignment whereas those of zinc and copper remain randomly oriented? A useful theory should also make quantitative predictions. For example, how does conductivity or magnetization vary as a function of temperature?
This last question brings us back to the Ising model, which was devised to explore the thermal properties of ferromagnets. If you heat a magnet to high temperature, the magnetization fades away. Then, when you allow the material to cool again, it regains its magnetic properties at a specific temperature called the Curie point (1,040 kelvins for iron). The transition is abrupt: A graph of magnetization as a function of temperature shows a discontinuity—a sharp kink—at the Curie point. Evidently there is a sudden transition from thermal chaos to magnetic order.
The Ising model was designed to capture the essential features of this behavior. The model was invented by the German physicist Wilhelm Lenz and investigated by his student Ernst Ising in the 1920s, at a time when the quantum theory of solids was still in its infancy. They represented the spin of an electron by a simple arrow constrained to point either up or down. Interactions of the spins were codified in two rules. First, pairs of adjacent spins prefer to point the same way, either both up or both down; there is an energy penalty whenever nearest-neighbor spins are antiparallel. Second, thermal fluctuations tend to mix up the spins, flipping them at random; thus orderly alignments are disrupted when the temperature rises.
Ising hoped to observe a sudden onset of magnetization, as in real ferromagnets. He analyzed a one-dimensional version of the model, in which the lattice is merely a line or a ring. Disappointingly, he found no discontinuous transition to a magnetized state at any temperature above absolute zero.
Ising believed that this negative result would carry over to higher dimensions as well, but a decade later other physicists found hints of magnetization in two dimensions. Then in 1944 Lars Onsager confirmed these results with an exact mathematical solution of the two-dimensional Ising model; his equation showed that magnetization in the planar spin system does indeed jump discontinuously at a nonzero critical temperature. For three dimensions no exact solution has ever been found, but computer simulations give unmistakable evidence of an abrupt phase transition.
The Ising model has gone on to become a kind of model for models. The same abstract structure—a lattice of sites, nearest-neighbor interactions, a variable at each site that takes on two discrete values—has served to describe not only magnets but also dozens of other physical systems, such as alloys (where the up and down spins represent atoms of two different elements) and gases (where the two states indicate the presence or absence of an atom). Ising-like models have even made their way into the social sciences, where they describe phenomena such as the emergence of racial segregation in housing patterns.
Meanwhile, despite all these diverse successes, the Ising model has not proved entirely satisfactory for its original purpose—as a tool for understanding ferromagnetism. In this application the model has two major weaknesses. First, spins in the Ising model are rigidly pinned to the lattice sites, but it turns out that some degree of electron mobility is crucial to many magnetic phenomena. Second, although the Ising model was inspired by quantum-mechanical ideas, it incorporates none of the peculiar rules and regulations that the quantum theory imposes on electrons. The Hubbard model addresses both of these issues.