In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining their microscopic properties.
The theory can also be given a general geometric setting, placing it in the context of Riemannian geometry, where in many cases exact solutions can be given.
of a superconductor near the superconducting transition can be expressed in terms of a complex order parameter field
is nonzero below a phase transition into a superconducting state, no direct interpretation of this parameter was given in the original paper.
and smallness of its gradients, the free energy density has the form of a field theory and exhibits U(1) gauge symmetry:
denotes the dissipation-free electric current density and Re the real part.
Consider a homogeneous superconductor where there is no superconducting current and the equation for ψ simplifies to:
Below the superconducting transition temperature, the above equation is expected to have a non-trivial solution (that is
: In Ginzburg–Landau theory the electrons that contribute to superconductivity were proposed to form a superfluid.
For T > Tc (normal phase), it is given by while for T < Tc (superconducting phase), where it is more relevant, it is given by It sets the exponential law according to which small perturbations of density of superconducting electrons recover their equilibrium value ψ0.
The penetration depth sets the exponential law according to which an external magnetic field decays inside the superconductor.
The phase transition from the normal state is of second order for Type II superconductors, taking into account fluctuations, as demonstrated by Dasgupta and Halperin, while for Type I superconductors it is of first order, as demonstrated by Halperin, Lubensky and Ma.
[4] In the original paper Ginzburg and Landau observed the existence of two types of superconductors depending on the energy of the interface between the normal and superconducting states.
The Meissner state breaks down when the applied magnetic field is too large.
In Type I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value Hc.
Depending on the geometry of the sample, one may obtain an intermediate state[5] consisting of a baroque pattern[6] of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field.
In Type II superconductors, raising the applied field past a critical value Hc1 leads to a mixed state (also known as the vortex state) in which an increasing amount of magnetic flux penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large.
He used Ginzburg–Landau theory to explain experiments on superconducting alloys and thin films.
[7] The Ginzburg–Landau functional can be formulated in the general setting of a complex vector bundle over a compact Riemannian manifold.
In multiple interesting cases, it can be shown to exhibit the same phenomena as the above, including Abrikosov vortices (see discussion below).
to be a one-form taking values in the Lie algebra corresponding to the symmetry group of the fiber.
generalizes the electromagnetic field strength to the non-Abelian setting, as the curvature form of an affine connection on a vector bundle .
To emphasize this, note that the first term of the Ginzburg–Landau functional, involving the field-strength only, is which is just the Yang–Mills action on a compact Riemannian manifold.
converge uniformly to zero, and the curvature becomes a sum over delta-function distributions at the vortices.
One achieves this by writing the exterior derivative as a sum of Dolbeault operators
The functional then becomes The integral is understood to be over the volume form so that is the total area of the surface
Integrating the second of these, one quickly finds that a non-trivial solution must obey Roughly speaking, this can be interpreted as an upper limit to the density of the Abrikosov vortecies.
The generalization to N = (2,2) supersymmetric theories in 2 spacetime dimensions was proposed by Cumrun Vafa and Nicholas Warner in November 1988;[17] in this generalization one imposes that the superpotential possess a degenerate critical point.
The same month, together with Brian Greene they argued that these theories are related by a renormalization group flow to sigma models on Calabi–Yau manifolds.
[20] Witten's sigma models were later used to describe the low energy dynamics of 4-dimensional gauge theories with monopoles as well as brane constructions.