Ginzburg–Landau theory
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In physics, Ginzburg–Landau theory, named after Vitaly Lazarevich Ginzburg and Lev Landau, is a mathematical physical theory used to describesuperconductivity. In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining their microscopic properties. Later, a version of Ginzburg–Landau theory was derived from the Bardeen-Cooper-Schrieffer microscopic theory by Lev Gor'kov, thus showing that it also appears in some limit of microscopic theory and giving microscopic interpretation of all its parameters.
Contents
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∙1Introduction
∙2Simple interpretation
∙3Coherence length and penetration depth
∙4Fluctuations in the Ginzburg–Landau model
∙5Classification of superconductors based on Ginzburg–Landau theory
∙6Landau–Ginzburg theories in string theory
∙7See also
∙8References
∙8.1Papers
Introduction[edit]
Based on Landau's previously-established theory of second-order phase transitions, Ginzburg and Landau argued that the free energy, F, of a superconductor near t
he superconducting transition can be expressed in terms of a complex order parameter field, ψ, which is nonzero below a phase transition into a superconducting state and is related to the density of the superconducting component, although no direct interpretation of this parameter was given in the original paper. Assuming smallness of |ψ| and smallness of its gradients, the free energy has the form of a field theory.
where Fn is the free energy in the normal phase, α and β in the initial argument were treated as phenomenological parameters, m is an effective mass, e is the charge of an electron, A is the magnetic vector potential, and  is the magnetic field. By minimizing the free energy with respect to variations in the order parameter and the vector potential, one arrives at the Ginzburg–Landau equations
where j denotes the dissipation-less electric current density and Re the real part. The first equation — which bears some similarities to the time-independentSchrödinger equation, but is principally different due to a nonlinear term — determines the order parameter, ψ. The second equation then provides the superconducting current.
Simple interpretation[edit]
Consider a homogeneous superconductor where there is no superconducting current and the equation for ψ simplifies to:
This equation has a trivial solution: ψ = 0. This corresponds to the normal state of the superconductor, that is for temperatures above the superconducting transition temperature, T>Tc.
Below the superconducting transition temperature, the above equation is expected to have a non-trivial solution (that is ψ ≠ 0). Under this assumption the equation above can be rearranged into:
When the right hand side of this equation is positive, there is a nonzero solution for ψ (remember that the magnitude of a complex number can be positive or zero). This can be achieved by assuming the following temperature dependence of α: α(T) = α0 (T - Tc) with α0 / β > 0:
∙Above the superconducting transition temperature, T > Tc, the expression α(T) / β is positive and the right hand side of the equation above is negative. The magnitude of a complex number must be a non-negative number, so only ψ = 0 solves the Ginzburg–Landau equation.
∙Below the superconducting transition temperature, T < Tc, the right hand side of the equation above is positive and there is a non-trivial solution for ψ. Furthermore
that is ψ approaches zero as T gets closer to Tc from below. Such a behaviour is typical for a second order phase transition.
In Ginzburg–Landau theory the electrons that contribute to superconductivity were proposed to form a superfluid.[1] In this interpretation, |ψ|2 indicates the fraction of electrons that have condensed into a superfluid.[1]
Coherence length and penetration depth[edit]
The Ginzburg–Landau equations predicted two new characteristic lengths in a superconductor which was termed coherence length, ξ. For T > Treaction diffusionc (normal phase), it is given by

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