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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.

## References

• M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Phys. Rev. B 40, 546 (1989).