Mexican-Hat Quench

A finite 48×48 periodic lattice illustrates relaxation of two real components under V=μ²r²+λr⁴, r²=φ₁²+φ₂². All quantities are dimensionless teaching units. For μ²≥0 the unique minimum is r=0, with equal Hessian eigenvalues 2μ². For μ²<0 the minima form a ring at v=√(−μ²/(2λ)); radial curvature is −4μ² and transverse curvature zero. Displayed curvature scales are their square roots, not measured Higgs-particle masses.

The overdamped rule is ∂tφ=0.35∇²φ−∇V+noise, with unit lattice spacing, periodic boundaries and Euler steps ≤0.02. μ² changes linearly over the quench time, then remains at its final value for three more simulation units. Initial components are uniform in ±noise amplitude. Independent uniform driving noise has amplitude×√dt and falls to 30% by the quench end. This is a seeded educational stochastic rule, not a calibrated thermal bath or relativistic field evolution. Zero noise and exactly zero initial field remain zero, even at an unstable origin.

Before a run, the preview compares the initial and final potentials; valid controls update this inexpensive view directly. Choose a preset and quench time, then Run to start the finite lattice. Parameters contains μ² endpoints, λ, noise and seed. Pause/Resume preserves the running state and fixed-step sequence. Hidden pages pause without catching up. Reset returns to the potential preview. Inputs lock during playback; edits while paused discard the old run. Only parameter choices and exact seed text are retained, including unfinished seed input. Fields, traces, progress and random-generator state are not saved; reload returns to the preview. Identical seed and parameters reproduce the field at the same simulation time. Progress ends at 100% after resuming too.

The field uses phase color and magnitude brightness relative to v when v>0. The radial curve is the instantaneous potential, with a red minimum. History plots the actual mean magnitude in green and instantaneous v as a dashed red line on a fixed time axis. A short noisy run need not reach v.

Phase regions are a heuristic count: sites above 0.35v are connected if neighboring phase differences are at most π/2, including wraparound edges. This is neither a defect count nor a thermodynamic observable; it is unavailable when v=0. The finite 2D animation does not demonstrate equilibrium long-range spontaneous order or evade the Mermin–Wagner theorem. A local mean-field potential minimum is distinct from a measured order parameter in an infinite system.

Tong, Continuous Symmetries §§4.1–4.2.