Ginzburg–Landau theory

In physics, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after Vitaly 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. One GL-type superconductor is the famous YBCO, and generally all Cuprates. 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. Based on Landau's previously established theory of second-order phase transitions, Ginzburg and Landau argued that the free energy density of a superconductor near the superconducting transition can be expressed in terms of a complex order parameter field , where the quantity is a measure of the local density of superconducting electrons analogous to a quantum mechanical wave function. While is nonzero below a phase transition into a superconducting state, no direct interpretation of this parameter was given in the original paper. Assuming smallness of and smallness of its gradients, the free energy density has the form of a field theory and exhibits U(1) gauge symmetry: where is the free energy density of the normal phase, and are phenomenological parameters, is an effective mass, is an effective charge (usually , where e is the charge of an electron), is the magnetic vector potential, and is the magnetic field. The total free energy is given by .