The Hubbard model captures the competition between electron kinetic energy (hopping t) and on-site Coulomb repulsion U. At large U/t, electrons localize — a Mott insulating phase emerges.
Hubbard Model: H = −t Σ cᵢ†cⱼ + U Σ nᵢ↑nᵢ↓ − μ Σ nᵢ. At half-filling (n=1), the Mott transition occurs near U/t ≈ 6 (Bethe lattice). Below: lower (LHB) and upper Hubbard bands (UHB) separated by the Mott gap Δ ≈ U − W, where W = bandwidth. The lattice visualization shows electron occupancy — doubly-occupied sites (blue) are energetically costly when U is large, driving localization (red = singly occupied, gray = empty).