Electrons dance to a quantum beat in the Hubbard model of solid-state physics.
Brian Hayes in American Scientist:
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.
A 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.