The Order Parameter
Ginzburg and Landau were playing a game that made mathematicians weep and physicists rejoice. It was 1950, and the microscopic theory didn't exist yet—BCS was seven years away—but they had intuition.
They started from observation: something changes at the transition temperature. Before, the material is normal. After, superconducting. There must be a quantity that is zero in the normal state and non-zero in the superconducting state. A quantity that orders the transition.
They called it the order parameter. Symbol: ψ (psi). It's a complex number at every point in space: ψ(r) = |ψ(r)| e^(iφ(r)). The amplitude |ψ| tells you the condensate density. The phase φ tells you the quantum coherence. In the normal state, ψ = 0. In the superconducting state, ψ ≠ 0. The system has chosen a phase. This is spontaneous symmetry breaking.
Ginzburg and Landau wrote down the free energy as a functional of ψ. Their energy functional had terms describing:
- The cost of spatial variations in ψ
- The interaction with the electromagnetic field (ψ carries charge 2e)
- A term proportional to |ψ|² that changes sign at Tc
- A term proportional to |ψ|⁴ that stabilizes the solution
The fourth-order term is the magic. Without it, the system collapses. With it, there's a minimum at non-zero |ψ|. The superconducting state is energetically favorable.
From this free energy, you derive the Ginzburg-Landau equations—two coupled differential equations. Replace the momentum operator with (−iℏ∇ − 2eA), and ψ's kinetic energy looks like that of a charged particle. The condensate behaves mathematically like a charged quantum fluid.
Two length scales emerge:
ξ (xi) — the coherence length. How far ψ can change before energy cost is prohibitive. λ (lambda) — the London penetration depth. How far a magnetic field penetrates.
The Ginzburg-Landau parameter: κ = λ/ξ. The ratio determines everything. κ < 1/√2 → Type I. κ > 1/√2 → Type II. Abrikosov later showed Type II superconductors allow quantized flux tubes, exactly as the Ginzburg-Landau equations predicted.
Abrikosov's work earned him the Nobel Prize in 2003, sharing with Ginzburg. Landau was long dead.
Ginzburg-Landau is not the full theory—BCS derives everything from first principles—but it is more useful in practice. It works near Tc, where BCS becomes intractable. It gives accurate predictions for vortex structures, interfaces, critical current. Engineers use Ginzburg-Landau every day building MRI magnets, even though they've never solved the BCS gap equation.
It's deeply connected to modern physics. The Ginzburg-Landau functional is mathematically identical to the Higgs mechanism in particle physics. The order parameter ψ is analogous to the Higgs field. Same mathematics, different scale.
Ginzburg navigated the Soviet regime, unable to attend the 1972 Nobel. He received the 2003 Nobel at age 89, attended, and died 13 days later.
Landau was legendary—brilliant, ruthless, beloved. His textbooks remain the standard reference. When he died in 1968, colleagues calculated his mental age as 300 years of physics knowledge.
Ginzburg and Landau guessed the right structure, wrote the right functional, and let mathematics do the rest.