Spontaneous Symmetry Breaking & Goldstone Modes

Landau Free Energy

Temperature T / T_c 1.40

Quartic stiffness b 0.60

Damping γ 0.35

Thermal fluctuations 0.10

Live State

Phase

Symmetric (T > T_c)

Order parameter |η|

0.00

Free energy F(η)

0.00

Radial mode ω_r

0.00

Goldstone mode ω_θ

0.00

How it works

The order parameter η = (η₁, η₂) of a system with continuous O(2) symmetry (a ferromagnet's magnetisation, a superconductor's gap, a superfluid's condensate phase) minimises the Landau free energy:

F(η) = a(T − T_c)|η|² + b|η|⁴,   |η|² = η₁² + η₂²

Above T_c the coefficient a(T−T_c) is positive, F has a single minimum at η = 0, and the ball sits at the centre of a plain bowl — the O(2) symmetry is unbroken and every direction is equally stiff.

Below T_c the quadratic term flips sign and the bowl inverts into a Mexican-hat / wine-bottle surface with a circular trough at radius

η₀ = √( a(T_c − T) / 2b )

The ball must roll off the now-unstable centre and settle somewhere on the trough — but the trough is perfectly flat, so which angle it picks is arbitrary. That random choice is spontaneous symmetry breaking: the ground state has less symmetry than the free energy itself.

Linearising the dynamics around a point on the trough splits into two independent normal modes:

Radial (amplitude) mode — restoring force along the wall of the trough, curvature k_r = 4a(T_c−T), frequency ω_r = 2√(a(T_c−T)). This is the "massive" mode (a Higgs-like excitation).
Angular (phase) mode — motion along the flat trough has zero restoring force, k_θ = 0, ω_θ = 0. This is the Goldstone mode: exactly massless, and it is why thermal noise makes the ball wander freely around the ring while barely leaving the valley floor radially.

The Kick radial / Kick angular buttons give the order parameter a small impulse along each direction so you can watch the radial mode ring down while an angular kick just relocates the ball with no restoring force at all. Damping sets how fast the radial ringing decays; Thermal fluctuations add Langevin noise every step, the same mechanism that lets a real order parameter's phase diffuse in a superconductor or superfluid.

Order parameter η Trajectory trail Trough (below T_c)

⚠ Couldn't load the 3D engineThree.js failed to load from the CDN. Check your connection and reload.