Lattice Gauge Theory: U(1) Plaquette Monte Carlo

Gauge Coupling

Coupling β = 1/g² 1.00

Wilson loop size R×R 2×2

MC sweeps / frame 6

Live Readouts

⟨plaquette action⟩

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Wilson loop W(R,R)

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Metropolis accept

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Sweeps done

How it works

A compact U(1) lattice gauge field assigns a phase θ_l ∈ [0, 2π) — a link variable U_l = e^iθ_l — to every edge of a 2D periodic lattice, instead of a value to every site. The gauge-invariant building block is the plaquette (elementary square loop):

θ_p = θ(x,y)ʰ + θ(x+1,y)ᵛ − θ(x,y+1)ʰ − θ(x,y)ᵛ
S[U] = β Σ_p (1 − cos θ_p),   β = 1/g²

The simulator thermalizes this action with a Metropolis Monte Carlo sweep: pick a random link, propose θ_l → θ_l + Δ, and accept the move with probability min(1, e^−ΔS), where ΔS only involves the two plaquettes that touch that link. Repeated sweeps drive the ensemble toward thermal equilibrium at the chosen β.

β slider — large β (weak coupling g) favors small, aligned plaquette phases (ordered, blue links); small β (strong coupling) drives disordered, high-flux plaquettes (bright orange).
Wilson loop R×R — the trace of link phases around a closed rectangle, W(R,R) = ⟨cos Σ_l∈∂C θ_l⟩, highlighted in lime on the lattice.
In 2D, compact U(1) gauge theory confines at every coupling: W decays as a strict area law, W ~ e^−σA, with string tension σ > 0. Watch W shrink as R grows or as β drops — the same area-law mechanism that confines quarks inside hadrons in real lattice QCD, just in the simplest possible dimension.

Real-world relevance: this Metropolis-on-link-variables scheme is the direct 2D ancestor of the lattice QCD Monte Carlo codes used to compute hadron masses and the quark-confinement string tension from first principles.

Ordered plaquette (low flux) Disordered plaquette (high flux) Wilson loop boundary

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