Bose-Hubbard: Mott–Superfluid Transition
Optical lattice bosons — competing hopping J vs on-site repulsion U
Bose-Hubbard Model: H = -J Σ(b†ᵢbⱼ + h.c.) + (U/2)Σnᵢ(nᵢ-1) - μΣnᵢ.
Superfluid phase (J≫U/z): bosons delocalized, phase coherent, compressible.
Mott insulator (U≫Jz): integer filling locked, gapped, incompressible.
Phase diagram shows Mott lobes in (J/U, μ/U) plane. Mean-field critical point: Jz/U = 1/(2n+1) for the n-th lobe.
Left: phase diagram with current point (red dot). Right: lattice occupation snapshot.