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In physics , Ginzburg—Landau theory , often called Landau—Ginzburg theory , named after Vitaly Lazarevich Ginzburg and Lev Landau , is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining their microscopic properties.

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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. 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.

This general setting then extends to quantum field theory and string theory , again owing to its solvability, and its close relation to other, similar systems. By minimizing the free energy with respect to variations in the order parameter and the vector potential, one arrives at the Ginzburg—Landau equations. The second equation then provides the superconducting current. Under this assumption the equation above can be rearranged into:. In Ginzburg—Landau theory the electrons that contribute to superconductivity were proposed to form a superfluid.

The Ginzburg—Landau equations predicted two new characteristic lengths in a superconductor. Thus this theory characterized all superconductors by two length scales. It was previously introduced by the London brothers in their London theory. Expressed in terms of the parameters of Ginzburg—Landau model it is. The penetration depth sets the exponential law according to which an external magnetic field decays inside the superconductor.

The original idea on the parameter "k" belongs to Landau. Taking into account fluctuations. For Type II superconductors, the phase transition from the normal state is of second order, as demonstrated by Dasgupta and Halperin. 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. Superconductors can be divided into two classes according to how this breakdown occurs.

Vortices in the Ginzburg-Landau model of superconductivity

In Type I superconductors , superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value H c. Depending on the geometry of the sample, one may obtain an intermediate state [2] consisting of a baroque pattern [3] 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 H c 1 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.

At a second critical field strength H c 2 , superconductivity is destroyed. The mixed state is actually caused by vortices in the electronic superfluid, sometimes called fluxons because the flux carried by these vortices is quantized. The most important finding from Ginzburg—Landau theory was made by Alexei Abrikosov in He used Ginzburg—Landau theory to explain experiments on superconducting alloys and thin films.

He found that in a type-II superconductor in a high magnetic field, the field penetrates in a triangular lattice of quantized tubes of flux vortices. 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. The Ginzburg—Landau functional is then a Lagrangian for that section:. The notation used here is as follows. The integral is explicitly over the volume form. It is conventionally written as.

Figure 5 Color Convergence of the bifurcation perturbation theory.

Figure 6 Feynman rules for vortex lattice. Figure 7 Two loops connected diagrams contributing to free energy. Figure 8 Feynman rules and the diagrams in the homogeneous phase: a propagator, b vertex, c the two loop energy correction.

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Figure 9 Color online Free energy in liquid. The curve T 0 is the Gaussian approximation, while T 1 ,… are higher order renormalized perturbation theory results.

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Optimized perturbation theory gives curves 1,2,… and finally BP lines are the Borel-Pade results. Figure 10 Additional diagram of the renormalized perturbation theory shown in a. Bubbles or cacti diagrams summed by the optimized expansion are shown in b — d. A diagram which is not of that type is shown in f.

Sylvia Serfaty (Author of Vortices in the Magnetic Ginzburg-Landau Model)

Figure 11 Color online The melting point and the spinodal point of the crystal. The free energies of the crystalline and the liquid states are equal at melt, while metastable crystal becomes unstable at spinodal point. Figure 13 Universal function d a T determining the shift of the melting line due to disorder. Figure 14 Color online Nb Se 2 phase diagram. Sign up to receive regular email alerts from Reviews of Modern Physics.

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Vortices in the Magnetic Ginzburg-Landau Model

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