Electrons dance to a quantum beat in the Hubbard model of solid-state physics
The Electron Do-Si-Do
The only exact solution of a Hubbard model applies to the one-dimensional case, where the electrons move back and forth along a line of sites. In 1968 Elliott H. Lieb, now of Princeton University, and F. Y. Wu of Northeastern University studied the behavior of the one-dimensional system at half-filling (N electrons on N sites) as the value of U is varied. At large U (strong repulsion), the system is an insulator; Lieb and Wu proved there is no transition to a conducting state at any U greater than zero. They also showed that the ground state of the linear model is not ferromagnetic but antiferromagnetic: The lowest-energy configuration is one in which alternate spins point in opposite directions.
Lieb and Wu got their results by exploiting special properties of one-dimensional systems. In particular, there is no way for one electron to exchange places with another electron except by performing a kind of do-si-do, in which the two particles simultaneously occupy the same site. Because of the Pauli principle, two electrons in the same spin state can never change their ordering along the line. These constraints, which simplify the analysis, do not hold in higher dimensions.
Forty years later, the work of Lieb and Wu remains the only rigorous solution to a Hubbard model. But there are lots of less-than-rigorous hints and clues from approximation methods and from computer simulations.
It is widely believed that the two- and three-dimensional models also have an antiferromagnetic ground state—a checkerboard of alternating up and down spins—when the lattice is half-filled. An informal argument in support of this view points out that a fully magnetized system has only one possible configuration, since none of the electrons can move; in the antiferromagnet, adjacent electrons with opposite spins can swap places through the do-si-do mechanism. Thus there are many equivalent configurations for an antiferromagnet, which lowers the overall energy.
Can a Hubbard model ever favor ferromagnetism? Yes. In the 1960s Yosuke Nagaoka of Kyoto University discovered a ferromagnetic phase that appears when two conditions are satisfied: the repulsive interaction U is very strong, and the number of electrons is just short of half-filled. A single vacancy (that is, N–1 electrons on N sites) is enough to make the difference! Ferromagnetism also emerges spontaneously in Hubbard-like models in which the electrons can hop farther than the nearest-neighbor sites. One such scheme was described in 1995 by Hal Tasaki of Gakushuin University in Japan.
What’s intriguing about both ferromagnetism and antiferromagnetism in Hubbard systems is that the orderly configurations arise even though the model includes no direct interactions between pairs of electrons that would tend to align the spins either parallel or antiparallel. This is quite different from the Ising model, where parallel spins benefit from an energy bonus. In the Hubbard model—and surely in real solids as well—long-range order comes from subtler correlations within the entire population of electrons.