Each lattice site holds up to two electrons (↑ and ↓). Electrons hop to empty same-spin neighbour sites at rate t; placing a second electron on an already-occupied site costs energy U.
Every attempted hop is accepted with Metropolis probability
min(1, e^(−ΔE / k_BT)), where ΔE = U·(Δ double-occupancy). As U/t grows, hops that would create a doubly-occupied site are exponentially suppressed — electrons freeze one-per-site and D collapses toward the Mott-insulating limit, even though nothing changed in the lattice geometry.
- U/t ≈ 0–2: electrons roam freely — metallic, D ≈ D₀.
- U/t ≈ 2–6: correlated metal — hopping still happens but double occupancy is suppressed below D₀.
- U/t ≳ 6 (near ⟨n⟩=1): Mott insulating regime — hopping that would double-occupy a site is almost always rejected, D → 0.
ΔE = U · [ n_other(j) − n_other(i) ]
P(accept) = 1 if ΔE ≤ 0
P(accept) = e^(−ΔE/k_BT) if ΔE > 0