 Bose–Hubbard model

A lattice boson Hamiltonian operator combining hopping and on-site repulsion. The on-site interaction counts distinct particle pairs through $n_i(n_i-1)/2$. It is the bosonic counterpart of the Hubbard model; projecting a continuum contact interaction onto localized orbitals gives $U=g\int |{\phi }_i{|}^4$ when intersite overlaps are negligible.

The Bose–Hubbard model is a theoretical framework used in condensed matter physics to describe the behavior of interacting bosons on a lattice. This model is particularly significant for studying phenomena such as superfluidity and the formation of Bose-Einstein condensates.