The Hubbard Model

One of the great successes of solid-state physics is the idea that many electrons in a crystal can often be described approximately as independent quasiparticles moving through an effective periodic potential.

This picture leads naturally to electronic band theory and explains the behavior of many metals, semiconductors, and insulators.

But some materials cannot be understood adequately this way. Electron–electron interactions can become strong enough that they determine the electronic state itself.

The Hubbard model was introduced as a minimal way to study this competition between electron motion and electron repulsion.

Despite its compact form, the Hubbard Hamiltonian contains the two ingredients needed to generate a wide range of correlated-electron behavior.

Much of the physics of the Hubbard model is controlled by the competition between \(t\) and \(U\), or more generally between the interaction strength and the electronic bandwidth.

If the interaction energy \(U\) is small compared with the kinetic energy scale, electrons can move relatively freely through the lattice.

In this regime, the system can often retain a metallic character and may be treated starting from an itinerant-electron picture.

Electron interactions are still present, but they act more as a correction to the motion of otherwise mobile quasiparticles.

If \(U\) becomes sufficiently large, putting two electrons on the same lattice site becomes energetically expensive.

An electron may then be prevented from hopping onto a neighboring occupied site even though ordinary band filling would otherwise permit electronic motion.

This interaction-driven localization is at the heart of Mott physics.

A particularly important case occurs when the lattice contains, on average, one electron per site. This is known as half filling for the single-band Hubbard model.

Without strong interactions, electrons could still hop between sites and form an electronic band.

But if the on-site repulsion is sufficiently strong, hopping onto an occupied site creates an energetically costly doubly occupied state. Electron motion can therefore become strongly suppressed.

A Mott insulator is one of the most important examples of a state where a simple noninteracting band description becomes inadequate.

The electrons are present at an electronic filling that might permit metallic behavior in a weakly interacting picture, yet strong electron–electron repulsion suppresses charge motion.

This is why the Mott state is fundamentally a many-body phenomenon.

Even when charge motion is strongly suppressed, the electron spins can still interact.

An electron can virtually hop onto a neighboring occupied site and return. These virtual processes generate an effective magnetic interaction known as superexchange.

In the simplest strong-coupling picture, this interaction favors antiferromagnetic alignment of neighboring spins.

Thus a model that begins only with hopping and on-site repulsion can naturally generate magnetic order.

The physics becomes even richer when electrons are added to or removed from a half-filled correlated state.

Empty sites or additional carriers provide new possibilities for charge motion while strong local interactions remain important.

The competition among carrier motion, magnetism, and correlations leads to a difficult many-body problem whose complete behavior is still not known analytically in two dimensions.

The Hubbard model became particularly important after the discovery of high-temperature superconductivity in copper-oxide materials.

The parent cuprates are strongly correlated antiferromagnetic insulators. When carriers are introduced through doping, they evolve into a much richer electronic phase diagram that includes superconductivity.

Because the two-dimensional Hubbard model contains both strong on-site interactions and antiferromagnetic correlations, it became one of the central minimal models used to investigate the underlying physics of the cuprates.

One reason the Hubbard model is so famous is the contrast between how easy it is to write down and how difficult it is to solve.

The many-electron Hilbert space grows extremely rapidly with system size, and interactions prevent the problem from being reduced to independent particles.

As a result, physicists use a wide range of analytical and numerical techniques, including exact diagonalization, quantum Monte Carlo, dynamical mean-field theory, density-matrix renormalization group, cluster methods, and tensor-network approaches.