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Hip-Hop Physics

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

Brian Hayes

An Electronic Mosh Pit

The Hubbard model was invented in the early 1960s by John Hubbard, a British physicist who died young in 1980. Martin Gutzwiller of IBM Research in Zurich devised a similar model at about the same time, and investigators in Japan were also thinking along the same lines.

There were two main motivations for the model—two phenomena in need of theoretical explanation. One aim was to understand the mechanism of ferromagnetism. There had certainly been progress in this direction since the time of Lenz and Ising; in particular, a model developed in the 1930s by Werner Heisenberg adopted the simple lattice of the Ising model but gave a more realistic quantum-mechanical account of how adjacent spins interact. Still, the Heisenberg model left the spins stationary on the lattice, whereas the electrons that give rise to ferromagnetism in elements such as iron and nickel are not strictly localized; they can migrate from atom to atom. The Hubbard model allowed for such motion.

The second question addressed by the Hubbard model concerned electrical conductivity—or the lack of it—in certain crystalline compounds such as copper oxide (CuO). In most insulators, all the electrons are tightly bound to atoms or molecules, leaving no mobile electrons to carry a current. In the case of CuO, the theory of solids suggested there should be an ample supply of conduction electrons, and yet the material is an insulator. In 1937 Nevill Mott proposed an explanation: CuO fails to conduct not for lack of electrons but because the electrons can’t get out of each other’s way. The conduction band of this substance is like a crowded dance floor, where everyone desperately wants to keep moving but there are no vacant spaces to move into. By the 1960s the Hubbard model offered hope of better understanding such Mott insulators.

In recent years, the struggle to understand new high-temperature superconducting materials has intensified interest in the Hubbard model. The superconductors are layered materials whose components include copper oxides. Philip W. Anderson of Princeton University has argued that a two-dimensional Hubbard model can account for the transition to superconductivity in the copper oxide layers. This view remains controversial; on the other hand, the mere possibility of solving the puzzle of cuprate superconductivity has led to a frenzy of work on the Hubbard model and its variations.

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