Bose-Hubbard model

From Wikipedia, the free encyclopedia

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.