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Hip-Hop Physics

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

Brian Hayes

The Crowded Dance Floor

The stage setting for the Hubbard model is the same as that of the Ising model: a simple lattice with cubic symmetry—a cartoon of a crystalline solid. But the Hubbard dancers are more acrobatic. As noted above, Hubbard electrons can jump from one lattice site to another. (The range of motion is usually limited to nearest-neighbor sites.) The electrons also interact with one another, experiencing mutual repulsion whenever two electrons land on the same site. Finally, the choreography of Hubbard electrons is subject to a special rule, the Pauli exclusion principle, a definitive element of quantum mechanics.

Think of the Pauli principle (named for the Austrian physicist Wolfgang Pauli) as a generalization of the commonsense notion that two objects cannot be in the same place at the same time. The quantum version says that no two particles can occupy exactly the same quantum state. If two electrons have the same energy, for example, they must differ in angular momentum or some other property. On the Hubbard lattice, the exclusion principle implies that if two electrons occupy the same site, they must have opposite spins. An obvious corollary is that no site can ever accommodate more than two electrons, since at least two of them would have the same spin.

With these facts in hand, we can get a rough vision of the Hubbard model in action. Suppose the lattice is two-dimensional, like a sheet of graph paper. Some of the lattice points are occupied by electrons; some of those electrons are spin-up and the rest are spin-down. Thus a site can have any of four occupation states: no electrons, one up electron, one down electron or a pair of electrons with opposite spins. An electron can hop to any neighboring site, provided the move is allowed by the exclusion principle.

There’s one more essential element to introduce: the energy of the electrons. The exclusion principle requires that electrons with the same spin have distinct energies, which means there must be a ladder of available energy levels. If all the electrons have the same spin, they will necessarily fill all the rungs of the ladder from bottom to top. However, if half the electrons are spin-up and half are spin-down, they can be packed two to a rung, lowering the average energy level. This sharing of levels means that configurations with mixed spins can be energetically favorable.

On the other hand, the presence of both up and down spins also allows pairs of electrons to occupy the same lattice site, which incurs an energy penalty because of their mutual repulsion. For each doubly occupied site, the overall energy of the system increases by an amount designated U. Thus there is a subtle competition between the cost of populating higher levels of the energy ladder and the cost of overcoming electromagnetic repulsion.

What happens when we push the Start button and let the electrons hop around on the lattice? In general, this is a very hard question, but a few “corner cases”—where some parameter is set to an extreme value—offer clues. One such parameter is the number of electrons. For a lattice of N sites, this number must lie between zero and 2N. Nothing much happens with zero electrons, of course, and it turns out the same is true with 2N electrons: All sites are filled with paired electrons, and none of the electrons can move.

Another parameter is U, the energy of electrostatic repulsion for electrons at the same lattice site. If U is zero (no repulsion at all), the spin-up and the spin-down electrons form two independent populations, each of which drifts through the lattice oblivious of the other’s existence. At the opposite extreme, if U is infinite, the repulsion is so great that no site ever holds more than one electron. In this circumstance electrons can move only when there is an adjacent vacant site; if the lattice is half full (N electrons, with no vacancies), the configuration is frozen solid.

Still another parameter, whose role I have neglected so far, is temperature. At a temperature of absolute zero, the Hubbard model is compelled to adopt the configuration of lowest possible energy—the ground state. Thermal agitation at higher temperatures allows the system to escape this fate. With warming, higher-energy states come within reach. At infinite temperature all possible configurations are equally likely, and energy differences between states cease to have any influence on the behavior of the system.

As a practical matter, interest focuses not on the extreme cases but on realistic values of the parameters. Physicists would most like to know what happens when the number of electrons is at or near half-filling (one electron per site) and when the repulsion parameter U is greater than zero but far from infinite. As for temperature, it is important to identify the ground state, but we would also like to know how the behavior of the system changes as it warms up from absolute zero.

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