The Hubbard model is an approximate model used, especially in solid-state physics, to describe the transition between conducting and insulating systems. The Hubbard model, named after John Hubbard, is a simple model of interacting particles in a lattice, with only two terms in the Hamiltonian (see example below): a kinetic term allowing for tunneling (“hopping”) of particles between sites of the lattice and a potential term consisting of an on-site interaction. The particles can either be fermions, as in Hubbard’s original work, or bosons, in which case the model is referred to as the “Bose–Hubbard model”.
The Hubbard model is a useful approximation for particles in a periodic potential at sufficiently low temperatures, where all the particles may be assumed to be in the lowest Bloch band, and long-range interactions between the particles can be ignored. If interactions between particles at different sites of the lattice are included, the model is often referred to as the “extended Hubbard model”. In particular, the Hubbard term, most commonly denoted by U, is applied in first principles based simulations using Density Functional Theory, DFT. The inclusion of the Hubbard term in DFT simulations is important as this improves the prediction of electron localisation and thus it prevents the incorrect prediction of metallic conduction in insulating systems.
The model was originally proposed in 1963 to describe electrons in solids. Since then, it has been applied to the study of high-temperature superconductivity, quantum magnetism, and charge density waves. The Hubbard model introduces short-range interactions between electrons to the tight-binding model, which only includes kinetic energy (a “hopping” term) and interactions with the atoms of the lattice (an “atomic” potential). When the interaction between electrons is strong, the behavior of the Hubbard model can be qualitatively different from a tight-binding model. For example, the Hubbard model correctly predicts the existence of Mott insulators: materials that are insulating due to the strong repulsion between electrons, even though they satisfy the usual criteria for conductors, such as having an odd number of electrons per unit cell.