| Ginzburg–Landau theory | |
|---|---|
| Name | Ginzburg–Landau theory |
| Caption | Phenomenological description of superconductivity and order parameters |
| Field | Condensed matter physics |
| Introduced | 1950 |
| Authors | Vitaly Ginzburg; Lev Landau |
| Notable concepts | Ginzburg–Landau free energy, order parameter, coherence length, penetration depth |
Ginzburg–Landau theory
Ginzburg–Landau theory is a phenomenological framework for describing phase transitions and spatially varying order in superconductors and other condensed matter systems. Developed to connect macroscopic electromagnetic behavior to a complex order parameter, it provides tractable partial differential equations that capture superconducting currents, vortices, and critical phenomena. The theory matters in Quantum Physics because it bridges microscopic quantum descriptions (e.g., BCS theory) and experimentally observed macroscopic quantum phenomena such as flux quantization and the Meissner effect.
Ginzburg–Landau theory was introduced in 1950 by Soviet physicists Vitaly Ginzburg and Lev Landau as an extension of Landau theory of phase transitions that includes spatial gradients and coupling to electromagnetic fields. It predates the microscopic explanation of superconductivity offered by Bardeen, Cooper, and Schrieffer in 1957. The phenomenological approach was influential in prompting quantitative comparisons with experiments at institutions such as the Kapitza Institute and laboratories across Europe and the United States. Subsequent work, notably by Gor'kov, derived the Ginzburg–Landau formalism from BCS theory near the critical temperature, establishing it as an asymptotic limit of microscopic many-body theory.
Central to the theory is the Ginzburg–Landau free energy functional, a Landau-type expansion in powers of a complex order parameter ψ(r). For a superconductor coupled to an electromagnetic vector potential A(r), the free energy density typically reads -α|ψ|^2 + (β/2)|ψ|^4 + (1/2m*)|(-iħ∇ - q*A)ψ|^2 + (|B|^2/2μ0), where α and β are phenomenological coefficients, m* and q* are effective mass and charge of the Cooper-pair condensate, and B = ∇×A is the magnetic induction. The coefficients α(T) and β are connected to critical behavior near the superconducting transition temperature Tc. The functional formalism makes contact with variational principles and permits derivation of Euler–Lagrange equations describing equilibrium configurations. Key experimental parameters such as the magnetic penetration depth and coherence length emerge from this free energy.
The complex scalar field ψ(r) serves as the order parameter representing macroscopic quantum coherence of the condensate. Its amplitude |ψ|^2 is proportional to condensate density, while its phase arg(ψ) encodes supercurrent flow and topological defects. The onset of nonzero ψ below Tc exemplifies spontaneous symmetry breaking of global U(1) gauge symmetry; coupling to the electromagnetic gauge field promotes the Anderson–Higgs mechanism in which the would-be Goldstone mode is absorbed, giving the photon an effective mass inside a superconductor. The phase dynamics relate to Josephson phenomena observed in Josephson junctions and to quantized vortices studied in Abrikosov vortex lattices.
Variation of the free energy yields the coupled Ginzburg–Landau equations: a nonlinear Schrödinger-type equation for ψ and a Maxwell equation for A with supercurrent source terms. Solutions include uniform superconducting states, one-dimensional domain walls, and localized vortex solutions characterized by phase winding and a normal core. In type II superconductors the theory predicts stable Abrikosov vortex lattices above the lower critical field Hc1 and below the upper critical field Hc2. Analytic and numerical methods—such as perturbation theory, variational ansätze, and finite-element simulations—are used to study vortex interactions, pinning by defects, and boundary effects in geometries relevant to experiments at facilities like CERN and national magnet laboratories.
Ginzburg–Landau theory introduces the dimensionless Ginzburg–Landau parameter κ = λ/ξ, the ratio of the magnetic penetration depth λ to the coherence length ξ. This parameter classifies superconductors: κ < 1/√2 (type I) exhibits macroscopic Meissner expulsion and first-order transitions; κ > 1/√2 (type II) supports mixed states with vortices. The coherence length measures spatial variations of the order parameter and connects to pair size in microscopic theories, while λ quantifies screening of magnetic fields. Experimental determinations of ξ and λ in materials such as niobium and high-temperature cuprates guide device design in quantum computing platforms and superconducting magnets.
Extensions of the static theory include the time-dependent Ginzburg–Landau equation (TDGL), used to model non-equilibrium dynamics, vortex motion, and response to time-varying fields; TDGL finds application in modeling flux-flow resistivity and relaxation phenomena. Microscopic derivations by Lev Gor'kov and subsequent work connect Ginzburg–Landau coefficients to parameters of BCS theory, the Eliashberg theory for strong coupling, and to multiband generalizations relevant for materials like MgB2 and iron-based superconductors. The formalism has been generalized to include anisotropy, unconventional pairing symmetries (d-wave, p-wave), and coupling to additional order parameters (charge-density waves, spin-density waves) studied in contexts such as cuprate superconductors and heavy-fermion compounds. Ginzburg–Landau–type approaches also inform theories of superfluidity in Helium-3, cold-atom condensates in BEC experiments, and effective field theories employed in condensed matter and high-energy physics, exemplifying its broad utility as a bridge between macroscopic observables and underlying quantum many-body phenomena.
Category:Superconductivity Category:Condensed matter physics