Superfluid Helium
Below 2.17 K, helium-4 undergoes a phase transition into a superfluid — a quantum state where viscosity vanishes entirely. Vorticity becomes quantized: rotation can only exist as discrete vortex filaments, each carrying exactly one quantum of circulation h/m. Click to inject vortex pairs; watch them orbit, scatter, and annihilate.
iℏ ∂ψ/∂t = −(ℏ²/2m)∇²ψ + g|ψ|²ψ — Gross–Pitaevskii equation
At Tλ = 2.1768 K, helium-4 transitions from a normal fluid (He I) to a superfluid (He II). Below this temperature, a macroscopic fraction of atoms condenses into the zero-momentum ground state — a Bose–Einstein condensate — giving the fluid its extraordinary properties.
In a superfluid, the velocity field derives from a quantum phase gradient: v = (ℏ/m)∇φ. This forces circulation to be quantized in units of κ = h/m ≈ 9.97 × 10⁻⁸ m²/s. Vortex cores have near-zero density — they are topological defects in the condensate wavefunction.
Landau's two-fluid model describes He II as a mixture of a superfluid component (zero entropy, zero viscosity) and a normal component (carries entropy, has viscosity). As temperature rises toward Tλ, the normal fraction grows and the superfluid fraction shrinks to zero.
The GPE is a nonlinear Schrödinger equation governing the condensate order parameter ψ. The |ψ|² term represents mean-field interactions between atoms. Vortex solutions have ψ → 0 at the core and a phase winding of 2π around each vortex, encoding quantized circulation.