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Hip-Hop Physics

Electrons dance to a quantum beat in the Hubbard model of solid-state physics

Brian Hayes

Mathematical models and computer simulations usually begin as aids to understanding, introduced when some aspect of natural science proves too knotty for direct analysis. Facing an intractable problem, we strip away all the messy details of the real world and build a toy universe, one simple enough that we can hope to master it. Often, though, even the dumbed-down model defies exact solution or accurate computation. Then the model itself becomes an object of scientific inquiry—a puzzle to be solved.

square-dancing electronsClick to Enlarge ImageA good example is the Ising model in solid-state physics, which attempts to explain the nature of magnetism in materials such as iron. (I wrote about the Ising model in an earlier Computing Science column; see “The World in a Spin,” September–October 2000.) The Ising model glosses over all the intricacies of atomic structure, representing a magnet as a simple array of electron “spins” on a plain, gridlike lattice. Even in this abstract form, however, the model presents serious challenges. Only a two-dimensional version has been solved exactly; for the three- dimensional model, getting accurate results requires both algorithmic sophistication and major computer power.

One step up from the Ising model—in terms of realism and complexity—is something called the Hubbard model. Again the aim is to describe aspects of solid-state physics, including various kinds of magnetism as well as certain conductive and insulating materials, and maybe even the high-temperature superconductors that have stumped theorists since the 1980s. As in the Ising model, the Hubbard model puts electrons on a simple lattice, but in this case the electrons are allowed to hop from site to site. The model also insists on a quantum-mechanical treatment of the interactions between electrons. These two features make the Hubbard model a much harder nut to crack.

Except in the special case of a one-dimensional lattice, the Hubbard model has defied exact mathematical analysis. And computer simulations of Hubbard systems become painfully slow with any more than a few dozen electrons. Calculations are so difficult that no one knows for sure whether various Hubbard systems are conductive or insulating, or what their magnetic properties might be. This situation has led to an extraordinary new strategy for solving the model: putting it to the test of experiment. Several groups of physicists have built macroscopic replicas of the Hubbard lattice out of light waves and trapped atoms. Thus we come full circle, creating a physical analog of an abstract model that in turn represents another physical system.

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