U(1) gauge field on 2D lattice — Wilson loop, plaquette action, Monte Carlo updates
Mean plaquette ⟨Re e^{iΦ_□}⟩: —
MC sweep: 0
Acceptance rate: —
Avg action S: —
Wilson action: S = −β Σ_□ Re[U_□] where U_□ = U_{x,μ}U_{x+μ,ν}U†_{x+ν,μ}U†_{x,ν}
For U(1): U = e^{iθ}, U_□ = e^{i(θ₁+θ₂−θ₃−θ₄)}
Wilson loop W(R,T) ∼ e^{−σRT} (confinement) or e^{−perimeter} (deconfinement).
β→∞: ordered phase; β→0: disordered. Phase transition at β_c≈1 in 2D compact U(1).