Hubbard Model

Physics: Hubbard Hamiltonian: H = −t∑c†c + U∑n↑n↓; t = hopping, U = on-site Coulomb repulsion. At half-filling (n=1) and U/t >> 1: Mott insulator — one electron per site, hopping costs energy U. Mott-Hubbard gap Δ ≈ U − W where W = 4t is bandwidth. Metal for U < U_c ≈ W (Brinkman-Rice). 1D exact solution via Bethe ansatz (Lieb-Wu 1968). Cuprate superconductors: half-filled Hubbard model on square lattice; holes doped away from half-filling become superconducting. Gutzwiller approximation: effective mass m*/m ~ 1/(1−(n/2)²) diverges at Mott transition.