Bose–Hubbard model

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 which 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 first 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 also be generalized and applied to Bose–Fermi mixtures, in which case the corresponding Hamiltonian is called the Bose–Fermi–Hubbard Hamiltonian.

The physics of this model is given by the Bose–Hubbard Hamiltonian:

${\displaystyle H=-t\sum _{\left\langle i,j\right\rangle }{\hat {b}}_{i}^{\dagger }{\hat {b}}_{j}+{\frac {U}{2}}\sum _{i}{\hat {n}}_{i}\left({\hat {n}}_{i}-1\right)-\mu \sum _{i}{\hat {n}}_{i}}$.

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