Each lattice site holds up to two electrons of opposite spin (Pauli exclusion). Every tick a random electron attempts to hop to a neighboring site:
H = -t Σ⟨ij⟩ c†ᵢcⱼ + U Σᵢ n↑ᵢn↓ᵢ
accept(hop) = 1 if target site empty
accept(hop) = e^(-U/t) if hop creates a doubly-occupied site
accept(hop) = 0 if Pauli-blocked (same spin already there)
At low U/t the Coulomb penalty barely matters — electrons hop almost freely between sites, double-occupancy flickers constantly and conductivity is high: a metal, exactly as independent-electron band theory predicts for a half-filled band.
At high U/t the energy cost of doubling up on a site becomes prohibitive. Electrons freeze one-per-site, hopping is suppressed exponentially, and conductivity collapses — a Mott insulator. Band theory alone cannot explain this: it takes electron-electron correlation, which is exactly what this Monte-Carlo-style acceptance rule captures. The crossover between the two regimes is the Mott transition.
- U/t slider — the ratio of on-site Coulomb repulsion U to hopping amplitude t; the single parameter that drives the metal-insulator transition.
- Start dynamics — runs the stochastic hopping process continuously.
- Reset — reseeds one electron per site (half filling) with random spins.