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Bose-Hubbard model

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:

H = -t \sum_{ \left\langle i, j \right\rangle } \left( b^{\dagger}_i b_j + b^{\dagger}_j b_i \right) + \frac{U}{2} \sum_{i} \hat{n}_i \left( \hat{n}_i - 1 \right) - \mu \sum_i \hat{n}_i.

Here i^{}_{} is summed over all lattice sites, and \left\langle i, j \right\rangle is summed over all neighboring sites. b^{\dagger}_i and b^{}_i are bosonic creation and annihilation operators. \hat{n}_i = b^{\dagger}_i b_i gives the number of particles on site i^{}_{}. t^{}_{} is the hopping matrix element, U^{}_{} > 0 is the on site repulsion, and \mu^{}_{} is the chemical potential.

At zero temperature the Bose Hubbard model (in the absence of disorder) is either in a Mott insulating (MI) state at small t \ or in a superfluid (SF) state at large t \. The Mott insulating phases is characterized by integer boson densities, by the existence of a gap for particle-hole excitations, and by zero compressibility. In the presence of disorder, a third, ‘‘Bose glass’’ phase exists. This phase is insulating because of the localization effects of the randomness. The Bose glass phase is characterized by a finite compressibility, the absence of a gap, and but an infinite superfluid susceptibility.



  • M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Phys. Rev. B 40, 546 (1989).
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Bose-Hubbard_model". A list of authors is available in Wikipedia.
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