Lattice Gauge Theory: U(1)

U(1) link variables: U_μ(x) = e^{iθ_μ(x)}
θ ∈ [−π, π]

Plaquette:
U_□ = U_x U_y U_x† U_y†

Wilson action (compact QED):
S = β Σ_□ (1 − Re U_□)

Metropolis: accept if
δS < 0 or r < e^{−δS}

Phase transition at β_c ≈ 1.01
(confinement ↔ deconfinement)

⟨P⟩ → 1 as β → ∞ (weak coupling)