Bose-Hubbard model
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The Bose-Hubbard model gives an approximate description of the physics of interacting bosons on a lattice. It is closely related to the Hubbard model which originated in solid state physics as an approximate description of the motion of electrons between the atoms of a crystalline solid. However, the Hubbard model applies to fermionic particles such as electrons, rather than bosons. The Bose-Hubbard model can be used to study systems such as bosonic atoms on an optical lattice.
The physics of this model is given by the Bose-Hubbard Hamiltonian:
.
Here 
is summed over all lattice sites, and 
is summed over all neighboring sites. 
and 
are bosonic creation and annihilation operators. 
gives the number of particles on site . 
is the hopping matrix element, 
is the on site repulsion, and 
is the chemical potential.
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