Bose-Hubbard model ($H=-{\sum }_{ij}{t}_{ij}{a}_i^{†}{a}_j+\frac{U}{2}{\sum }_i{n}_i(n_i-1)-\mu {\sum }_i{n}_i$)

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.

Two localized orbitals coupled by quantum tunnelling and on-site interactions give the smallest Bose-Hubbard model. A symmetric double-well potential with lowest one-particle energies $E_0,E_1$ has $t=(E_1-E_0)/2$ and common diagonal energy $(E_0+E_1)/2$. At fixed total particle number, that common diagonal term is constant. A fixed-number two-mode condensate gives a simple variational method for this model, while stronger interactions can require occupation-correlated trial quantum states.