Z₂ Gauge Theory — Confinement & Deconfinement
Z₂ lattice gauge theory on a 2D square lattice. Gauge variables σ=±1 live on links. Wilson loop ⟨W_C⟩ obeys area law (confined) at weak coupling and perimeter law (deconfined) at strong coupling. Monte Carlo with Metropolis updates.
Physics: H = -β Σ_□ σ₁σ₂σ₃σ₄ (product around each plaquette). Elitzur's theorem: local gauge symmetry cannot break spontaneously — only gauge-invariant observables are meaningful. The confinement-deconfinement transition (β_c≈0.44 for Z₂ in 2D) maps exactly to the 2D Ising transition via Kramers-Wannier duality. In the confined phase β<β_c: ⟨W⟩~e^{-σA}, string tension σ>0. In deconfined phase β>β_c: ⟨W⟩~e^{-μP}, only perimeter contribution.