Hubbard Model

The Hubbard model is the drosophila of condensed matter physics. It is perhaps the simplest possible model capturing the competition between localization of electrons in solids due to Coulomb repulsion and delocalization in energy bands due to kinetic energy lowering. Invented in the early 1960s to descibe magnetism in transition-metal monoxides, it has been generalized and applied to a host of problems in condensed matter including heavy fermions or high-temperature superconductors. Despite its apparent simplicity it shows complicated phase diagrams that depend on dimensionality and lattice coordination as well as electronic filling, with only few exact solutions in limiting cases known to this date.