Live lattice slice — plaquette action density 1−cos θ_P
low energyhigh energy
loop outline
W(R,T)
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V(R) = −ln W(R,T+1)/W(R,T)
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χ(R,T) string tension
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β/2 check (strong cpl.)
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Physics notes & correctness checks
• Action: S = β·Σ_P [1 − cos θ_P], plaquette phase θ_P = θ_μ(n)+θ_ν(n+μ̂)−θ_μ(n+ν̂)−θ_ν(n) (counterclockwise product).
• ΔS per Metropolis move depends only on the 4 staples touching the link (2 directions × forward+backward), never a full-action recompute.
• Strong-coupling check: at small β, ⟨cosθ_P⟩ → β/2 (shown as the dashed overlay on the scan plot). At large β, ⟨cosθ_P⟩ → 1−1/(3β) (free-field weak-coupling, d=3).
• Confinement: 2+1D compact U(1) confines for all β (Polyakov). The Creutz ratio χ(R,T) = −ln[W(R+1,T+1)W(R,T)/(W(R+1,T)W(R,T+1))] → string tension σ, nonzero and decreasing with β. This is emergent from the geometry/dynamics — nothing is hardcoded.
• Auto-tune holds Metropolis acceptance near ~50%; the equilibration plateau appears in the history plot.
• ΔS per Metropolis move depends only on the 4 staples touching the link (2 directions × forward+backward), never a full-action recompute.
• Strong-coupling check: at small β, ⟨cosθ_P⟩ → β/2 (shown as the dashed overlay on the scan plot). At large β, ⟨cosθ_P⟩ → 1−1/(3β) (free-field weak-coupling, d=3).
• Confinement: 2+1D compact U(1) confines for all β (Polyakov). The Creutz ratio χ(R,T) = −ln[W(R+1,T+1)W(R,T)/(W(R+1,T)W(R,T+1))] → string tension σ, nonzero and decreasing with β. This is emergent from the geometry/dynamics — nothing is hardcoded.
• Auto-tune holds Metropolis acceptance near ~50%; the equilibration plateau appears in the history plot.