Compact U(1) — 2+1D

Pure gauge Monte Carlo · Wilson action

Lattice & coupling

Lattice L8

beta = 1/g²1.00

step eps1.00

auto-tune eps (~50%)

update

Sweeps

thermalization

measurement

meas interval

β scan (confinement)

β min

β max

steps

Scan re-thermalizes at each β (warm start), then averages ⟨cosθ_P⟩ and the Creutz ratios. Headline confinement plot.

View

z-slice (time)0

Wilson R2

Wilson T2

sweep

acceptance

—

⟨cos θ_P⟩

—

sweeps / sec

—

Idle. Press Start to thermalize and measure at the current β, or Run β scan for the confinement curve.

Live lattice slice — plaquette action density (1 − cos θ_P)

low

high

Plane (x,y) at fixed z. Each cell = one elementary plaquette, colored by its local action (normalized to the slice). The yellow rectangle is the selected R×T Wilson loop.

selected loop

2×2

⟨W(R,T)⟩ measured

—

implied V(R) = −ln W / T

—

Wilson loops are measured after thermalization.

(a) Monte Carlo history — ⟨cos θ_P⟩ vs sweep

(b) β scan — ⟨cos θ_P⟩ vs β (overlays: β/2, I₁/I₀)

(c) Static potential V(R) vs R

(d) String tension — Creutz ratio χ(R,T) vs β

Confinement signature (2+1D compact U(1)): χ stays positive for all β and decreases with β. The β/2 line in (b) is the strong-coupling check — the measured points must hug it at small β.