Ginzburg-Landau theory

Ginzburg-Landau theory is a phenomenological framework for describing phase transitions and pattern formation through an order parameter whose dynamics are governed by a complex partial differential equation. Originally developed for superconductivity — where the order parameter represents the Cooper pair condensate — the theory has proven to be far more general, describing everything from stripe formation in pattern-forming systems to the onset of superfluidity in helium.

The Ginzburg-Landau equation is not merely a mathematical convenience. It is the universal amplitude equation for systems with a complex order parameter near a continuous bifurcation, emerging from symmetry principles rather than from microscopic physics. In two dimensions, the equation predicts the existence of Abrikosov vortices — topological defects that organize into regular lattices and whose dynamics determine the transport properties of type-II superconductors.