Bose–Hubbard model

The Bose–Hubbard model gives a description of the physics of interacting spinless bosons on a lattice. It is closely related to the Hubbard model that originated in solid-state physics as an approximate description of superconducting systems and the motion of electrons between the atoms of a crystalline solid. The model was introduced by Gersch and Knollman in 1963 in the context of granular superconductors. (The term 'Bose' in its name refers to the fact that the particles in the system are bosonic.) The model rose to prominence in the 1980s after it was found to capture the essence of the superfluid-insulator transition in a way that was much more mathematically tractable than fermionic metal-insulator models. The Bose–Hubbard model can be used to describe physical systems such as bosonic atoms in an optical lattice, as well as certain magnetic insulators. Furthermore, it can be generalized and applied to Bose–Fermi mixtures, in which case the corresponding Hamiltonian is called the Bose–Fermi–Hubbard Hamiltonian. The physics of this model is given by the Bose–Hubbard Hamiltonian: Here, denotes summation over all neighboring lattice sites and , while and are bosonic creation and annihilation operators such that gives the number of particles on site . The model is parametrized by the hopping amplitude that describes boson mobility in the lattice, the on-site interaction which can be attractive () or repulsive (), and the chemical potential , which essentially sets the number of particles. If unspecified, typically the phrase 'Bose–Hubbard model' refers to the case where the on-site interaction is repulsive. This Hamiltonian has a global symmetry, which means that it is invariant (its physical properties are unchanged) by the transformation . In a superfluid phase, this symmetry is spontaneously broken. The dimension of the Hilbert space of the Bose–Hubbard model is given by , where is the total number of particles, while denotes the total number of lattice sites.