Hubbard model ( from Wikipedia )
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The Hubbard model is an approximate model used, especially in solid state physics, to describe the transition between conducting and insulating systems.[1] The Hubbard model, named after John Hubbard, is the simplest 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, when the model is referred to as either the 'Bose–Hubbard model' or the 'boson Hubbard model'.
The Hubbard model is a good approximation for particles in a periodic potential at sufficiently low temperatures that all the particles are in the lowest Bloch band, as long as any long-range interactions between the particles can be ignored. If interactions between particles on different sites of the lattice are included, the model is often referred to as the 'extended Hubbard model'.
The model was originally proposed (in 1963) to describe electrons in solids and has since been the focus of particular interest as a model for high-temperature superconductivity. More recently, the Bose–Hubbard model has been used to describe the behavior of ultracold atoms trapped in optical lattices. Recent ultracold atom experiments have also realised the original, fermionic Hubbard model in the hope that such experiments could yield its phase diagram.[2]
For electrons in a solid, the Hubbard model can be considered as an improvement on the tight-binding model, which includes only the hopping term. For strong interactions, it can give qualitatively different behavior from the tight-binding model, and correctly predicts the existence of so-called Mott insulators, which are prevented from becoming conducting by the strong repulsion between the particles.
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