Mott insulator - 2D exact Hubbard cluster

The Hubbard model

The Hubbard Hamiltonian H = -t Σ_⟨ij⟩,σ (c^†_iσ c_jσ + h.c.) + U Σ_i n_i↑ n_i↓ has two competing terms: hopping t lowers the energy by spreading electrons over the lattice, and the on-site repulsion U penalises two electrons on one site. This page builds the full many-electron Hamiltonian of a small ring or chain at half filling (one electron per site on average, equal numbers of up and down spins) and finds its exact ground state with the Lanczos method. There is no sampling and no random hopping rule.

What it shows

The double occupancy D = ⟨n_↑n_↓⟩ per site falls from 1/4 at U = 0 towards zero as U/t grows, and the local moment ⟨(n_↑ - n_↓)²⟩ = 1 - 2D rises to 1: each site holds a single electron of definite spin. The charge gap Δ = E(N+1) + E(N-1) - 2E(N) (the cost of adding plus removing one electron) opens up, growing roughly as U minus the bandwidth. The Hellmann-Feynman theorem links the two: dE₀/dU = ⟨Σ n_↑n_↓⟩.

Honest simplifications: a finite cluster has no sharp phase transition; what you see is a smooth crossover. In a one-dimensional chain (Lieb and Wu) the infinite system is insulating for any U > 0, whereas the metal-insulator transition at a finite U/t in two and three dimensions needs larger lattices than can be solved exactly here. Rings with 4 or 8 sites have a degenerate ground state at U = 0, so the observables at very small U can depend on the state the solver picks; 2 and 6 sites are clean. The 3D companion uses a simple stochastic hopping rule instead of this exact solution.

• Drag U / t from 0 to 12 and watch the double occupancy curve fall and the local moment curve rise.
• Compare a ring with an open chain at the same size: the gap and the energy per site differ at the edges.
• Choose 2 sites and compare the exact numbers with the closed-form dimer solution quoted in the checks.
• Look at the charge gap curve: at small U it stays small, then grows almost linearly with U.