The Hubbard Model : Its Physics and Mathematical Physics

In the slightly more than thirty years since its formulation, the Hubbard model has become a central component of modern many-body physics.

It provides a paradigm for strongly correlated, interacting electronic systems and offers insights not only into the general underlying mathematical structure of many-body systems but also into the experimental behavior of many novel electronic materials.

In condensed matter physics, the Hubbard model represents the simplest theoret­ ical framework for describing interacting electrons in a crystal lattice.

Containing only two explicit parameters - the ratio ("Ujt") between the Coulomb repulsion and the kinetic energy of the electrons, and the filling (p) of the available electronic band - and one implicit parameter - the structure of the underlying lattice - it appears nonetheless capable of capturing behavior ranging from metallic to insulating and from magnetism to superconductivity.

Introduced originally as a model of magnetism of transition met­ als, the Hubbard model has seen a spectacular recent renaissance in connection with possible applications to high-Tc superconductivity, for which particular emphasis has been placed on the phase diagram of the two-dimensional variant of the model.

In mathematical physics, the Hubbard model has also had an essential role.

The solution by Lieb and Wu of the one-dimensional Hubbard model by Bethe Ansatz provided the stimulus for a broad and continuing effort to study "solvable" many-body models.

In higher dimensions, there have been important but isolated exact results (e. g. , N agoaka's Theorem).