Spontaneous Symmetry Breaking & Goldstone Modes

Ferromagnets, superconductors and superfluids all share the same mathematical skeleton: a Landau free energy F(η) written in a rotationally symmetric order parameter η, whose shape flips from a single bowl to a Mexican-hat trough as the system cools through a critical temperature T_c. This simulator renders that free-energy surface in real 3D and drops a damped, noise-driven particle onto it, so you can cool the system, watch the trough form, and see the order parameter spontaneously settle at a random angle — breaking a symmetry the underlying physics never had a preference against. Live readouts split the resulting dynamics into its two normal modes: a massive radial (amplitude) mode with a real restoring force and a genuinely massless angular Goldstone mode that only diffuses, exactly as Goldstone's theorem predicts for any continuous symmetry broken this way.