This is a spatially-extended companion to the 0D "ball on a Mexican hat" toy: instead of one order parameter η, every site of an N×N lattice carries its own vector η(x) = (η₁,η₂), and each site's local double-well potential is now coupled to its four neighbours, giving the real (discretised) Ginzburg-Landau free-energy functional:
F = Σ_sites [ a(T−Tc)|η|² + b|η|⁴ ] + Σ_bonds (κ/2)|η_i − η_j|²
The field relaxes by pure gradient descent on F (time-dependent Ginzburg-Landau / "Model A" dynamics — first-order in time, no inertia, standard for a non-conserved order parameter near a critical point) plus Langevin thermal noise:
γ ∂η/∂t = −2(a(T−Tc) + 2b|η|²)η + κ∇²η + noise(t)
Cool the lattice below Tc (a(T−Tc) < 0) and each patch of the field rolls off the now-unstable centre onto the circular trough — but different patches pick different, uncorrelated angles at first. The κ∇²η coupling then makes neighbouring patches drag each other toward agreement, so domains of shared angle nucleate and coarsen over time — this is the spatial mechanism behind spontaneous symmetry breaking that a single 0D particle cannot show at all.
Linearising around an ordered patch splits the dynamics into the same two normal modes as the 0D model, but now as genuine dispersion relations over wavevector k:
- Amplitude (radial) mode — ω_r(k) = 2√(a(Tc−T)) + κk². Gapped: even at k=0 it costs energy to change |η|, exactly matching the 0D toy's radial frequency.
- Goldstone (phase) mode — ω_θ(k) = κk². Exactly zero at k=0 — rotating every site's η by the same angle costs nothing, since a rigid rotation changes neither |η_i| nor |η_i−η_j| — but genuinely nonzero, and controllable by κ, at any finite k. This k-dependence is the real content of Goldstone's theorem that a single point-particle model cannot represent, because it has no space for a wave to propagate through.
Because the order parameter lives in 2D space, it also supports topological defects that are impossible in 0D: vortices, points where the phase winds by a full ±2π on a loop around them and the amplitude |η| must vanish at the core. Seed vortex pair writes in an exact vortex–antivortex configuration; Vortex pairs in the live stats is computed every frame from the standard winding-number sum around every elementary plaquette. Kick amplitude nudges every site's |η| uniformly (relaxes back at the gapped rate ω_r); Kick Goldstone rotates every site by the same random angle (a genuinely zero-cost move — watch the free energy not move at all).