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.
Hamiltonian
The physics of this model is given by the Bose–Hubbard Hamiltonian:
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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.
Hilbert space
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. At fixed or , the Hilbert space dimension grows polynomially, but at a fixed density of bosons per site, it grows exponentially as . Analogous Hamiltonians may be formulated to describe spinless fermions (the Fermi-Hubbard model) or mixtures of different atom species (Bose–Fermi mixtures, for example). In the case of a mixture, the Hilbert space is simply the tensor product of the Hilbert spaces of the individual species. Typically additional terms are included to model interaction between species.
Phase diagram
At zero temperature, the Bose–Hubbard model (in the absence of disorder) is in either a Mott insulating state at small , or in a superfluid state at large . The Mott insulating phases are characterized by integer boson densities, by the existence of an energy gap for particle-hole excitations, and by zero compressibility. The superfluid is characterized by long-range phase coherence, a spontaneous breaking of the Hamiltonian's continuous symmetry, a non-zero compressibility and superfluid susceptibility. At non-zero temperature, in certain parameter regimes a regular fluid phase appears that does not break the symmetry and does not display phase coherence. Both of these phases have been experimentally observed in ultracold atomic gases.
In the presence of disorder, a third, "Bose glass" phase exists. The Bose glass is a Griffiths phase, and can be thought of as a Mott insulator containing rare 'puddles' of superfluid. These superfluid pools are not interconnected, so the system remains insulating, but their presence significantly changes model thermodynamics. The Bose glass phase is characterized by finite compressibility, the absence of a gap, and by an infinite superfluid susceptibility. It is insulating despite the absence of a gap, as low tunneling prevents the generation of excitations which, although close in energy, are spatially separated. The Bose glass has a non-zero Edwards–Anderson order parameter and has been suggested (but not proven) to display replica symmetry breaking.