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. Alexei Abrikosov later used the theory to describe type-II superconductors as well. One GL-type superconductor is 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 a limit of the microscopic theory and giving it a 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. Ginzburg was awarded a third of the 2003 Nobel prize in physics for his development of the theory. Abrikosov was awarded another third of this prize for description of type-II superconductors using the theory.
Introduction
Free energy Based on Landau's previously established theory of second-order phase transitions Ginzburg and Landau argued that the free energy density f s {\displaystyle f_{s}} of a superconductor near the superconducting transition can be expressed in terms of a complex order parameter field ψ ( r ) = | ψ ( r ) | e i ϕ ( r ) {\displaystyle \psi (r)=|\psi (r)|e^{i\phi (r)}} , where the quantity | ψ ( r ) | 2 {\displaystyle |\psi (r)|^{2}} is a measure of the local density of superconducting electrons n s ( r ) {\displaystyle n_{s}(r)} analogous to a quantum mechanical wave function. In the convention used in this article, the relation is | ψ ( r ) | 2 = n s ( r ) / 2 {\displaystyle |\psi (r)|^{2}=n_{s}(r)/2} , which is the density of Cooper pairs. While ψ ( r ) {\displaystyle \psi (r)} 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 | ψ | {\displaystyle |\psi |} and smallness of its gradients, the free energy density has the form of a field theory and exhibits U(1) gauge symmetry. In Gaussian units, it is written as
f s = f n + α ( T ) | ψ | 2 + 1 2 β ( T ) | ψ | 4 + 1 4 m | ( − i ℏ ∇ − 2 e c A ) ψ | 2 + H 2 8 π , {\displaystyle f_{s}=f_{n}+\alpha (T)|\psi |^{2}+{\frac {1}{2}}\beta (T)|\psi |^{4}+{\frac {1}{4m}}\left|\left(-i\hbar \nabla -{\frac {2e}{c}}\mathbf {A} \right)\psi \right|^{2}+{\frac {\mathbf {H} ^{2}}{8\pi }},}
where
f n {\displaystyle f_{n}} is the free energy density of the normal phase,
α ( T ) {\displaystyle \alpha (T)} and β ( T ) {\displaystyle \beta (T)} are phenomenological parameters that are functions of T {\displaystyle T} (and often written just α {\displaystyle \alpha } and β {\displaystyle \beta } ).
2 m {\displaystyle 2m} the mass of a Cooper pair,
2 e {\displaystyle 2e} is the charge of a Cooper pair,
A {\displaystyle \mathbf {A} } is the magnetic vector potential, and
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