BoseHubbard

The Bose-Hubbard model is a theoretical framework used to describe interacting bosons on a lattice. It captures the competition between kinetic energy from hopping between neighboring sites, on-site repulsive interactions, and the chemical potential that fixes particle number.

The model is defined by the Hamiltonian H = -t Σ⟨i,j⟩ (a_i† a_j) + (U/2) Σ_i n_i(n_i-1) - μ Σ_i n_i, where a_i† and a_i are boson creation and annihilation operators at lattice site i, n_i = a_i† a_i is the number operator, t is the nearest-neighbor hopping amplitude, U is the on-site interaction strength, and μ is the chemical potential. The sum ⟨i,j⟩ runs over neighboring lattice pairs, and the model assumes a fixed lattice geometry with identical sites.

In the ground state, the system can exhibit a superfluid phase for large t/U, characterized by long-range phase coherence and gapless excitations. For large U/t at integer filling, it can enter a Mott insulator phase with a fixed integer number of bosons per site, a finite energy gap, and no long-range coherence. The zero-temperature phase diagram features Mott-insulating lobes surrounded by superfluid regions; the transition is a quantum phase transition driven by t/U and μ.

Experimental realizations include ultracold atomic gases in optical lattices, where lattice depth tunes t and U and thereby controls the phase. The Bose-Hubbard model serves as a basis for numerous numerical studies, using methods such as quantum Monte Carlo and density-matrix renormalization group, and it has inspired various extensions, including disorder-induced Bose glasses, extended models with nearest-neighbor interactions, and multi-component or spinor boson variants.