Ginzburg–Landau theory

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. 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.

Introduction

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 the 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.

F=F_{n}+\alpha |\psi |^{2}+{\frac {\beta }{2}}|\psi |^{4}+{\frac {1}{2m}}\left|\left(-i\hbar \nabla -2e\mathbf {A} \right)\psi \right|^{2}+{\frac {|\mathbf {B} |^{2}}{2\mu _{0}}}

where F_n 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 $\mathbf {B} =\nabla \times \mathbf {A}$ 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

\alpha \psi +\beta |\psi |^{2}\psi +{\frac {1}{2m}}\left(-i\hbar \nabla -2e\mathbf {A} \right)^{2}\psi =0

\nabla \times \mathbf {B} =\mu _{0}\mathbf {j} \;\;;\;\;\mathbf {j} ={\frac {2e}{m}}\operatorname {Re} \left\{\psi ^{*}\left(-i\hbar \nabla -2e\mathbf {A} \right)\psi \right\}

where j denotes the dissipation-less electric current density and Re the real part. The first equation — which bears some similarities to the time-independent Schrödinger 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

Consider a homogeneous superconductor where there is no superconducting current and the equation for ψ simplifies to:

\alpha \psi +\beta |\psi |^{2}\psi =0.\,

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 > T_c.

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:

|\psi |^{2}=-{\frac {\alpha }{\beta }}.

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 − T_c) with α₀/β > 0:

Above the superconducting transition temperature, T > T_c, 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 < T_c, the right hand side of the equation above is positive and there is a non-trivial solution for ψ. Furthermore,

|\psi |^{2}=-{\frac {\alpha _{0}(T-T_{c})}{\beta }},

that is ψ approaches zero as T gets closer to T_c 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

The Ginzburg–Landau equations predicted two new characteristic lengths in a superconductor which was termed coherence length, ξ. For T > T_c (normal phase), it is given by

\xi ={\sqrt {\frac {\hbar ^{2}}{2m|\alpha |}}}.

while for T < T_c (superconducting phase), where it is more relevant, it is given by

\xi ={\sqrt {\frac {\hbar ^{2}}{4m|\alpha |}}}.

It sets the exponential law according to which small perturbations of density of superconducting electrons recover their equilibrium value ψ₀. Thus this theory characterized all superconductors by two length scales. The second one is the penetration depth, λ. It was previously introduced by the London brothers in their London theory. Expressed in terms of the parameters of Ginzburg–Landau model it is

\lambda ={\sqrt {\frac {m}{4\mu _{0}e^{2}\psi _{0}^{2}}}},

where ψ₀ is the equilibrium value of the order parameter in the absence of an electromagnetic field. 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. The ratio κ = λ/ξ is presently known as the Ginzburg–Landau parameter. It has been proposed by Landau that Type I superconductors are those with 0 < κ < 1/√2, and Type II superconductors those with κ > 1/√2.

The exponential decay of the magnetic field is equivalent with the Higgs mechanism in high-energy physics.

Fluctuations in the Ginzburg–Landau model

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. While for Type I superconductors it is of first order as demonstrated by Halperin, Lubensky and Ma.

Classification of superconductors based on Ginzburg–Landau theory

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. 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_c1 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_c2, 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. Most pure elemental superconductors, except niobium and carbon nanotubes, are Type I, while almost all impure and compound superconductors are Type II.

The most important finding from Ginzburg–Landau theory was made by Alexei Abrikosov in 1957. 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.^[4]

Landau–Ginzburg theories in string theory

In particle physics, any quantum field theory with a unique classical vacuum state and a potential energy with a degenerate critical point is called a Landau–Ginzburg theory. The generalization to N = (2,2) supersymmetric theories in 2 spacetime dimensions was proposed by Cumrun Vafa and Nicholas Warner in the November 1988 article Catastrophes and the Classification of Conformal Theories, 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 in the paper Calabi–Yau Manifolds and Renormalization Group Flows. In his 1993 paper Phases of N = 2 theories in two-dimensions, Edward Witten argued that Landau–Ginzburg theories and sigma models on Calabi–Yau manifolds are different phases of the same theory. A construction of such a duality was given by relating the Gromov–Witten theory of Calabi–Yau orbifolds to FJRW theory an analogous Landau–Ginzburg "FJRW" theory in The Witten Equation, Mirror Symmetry and Quantum Singularity Theory. 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. Gaiotto, Gukov & Seiberg (2013)

References

1 2 Ginzburg VL (July 2004). "On superconductivity and superfluidity (what I have and have not managed to do), as well as on the 'physical minimum' at the beginning of the 21 st century". Chemphyschem. 5 (7): 930–945. PMID 15298379. doi:10.1002/cphc.200400182.
↑ Lev D. Landau; Evgeny M. Lifschitz (1984). Electrodynamics of Continuous Media. Course of Theoretical Physics. 8. Oxford: Butterworth-Heinemann. ISBN 0-7506-2634-8.
↑ David J. E. Callaway (1990). "On the remarkable structure of the superconducting intermediate state". Nuclear Physics B. 344 (3): 627–645. Bibcode:1990NuPhB.344..627C. doi:10.1016/0550-3213(90)90672-Z.
↑ Abrikosov, A. A. (1957). The magnetic properties of superconducting alloys. Journal of Physics and Chemistry of Solids, 2(3), 199–208.

General references

Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag ISBN 3-540-42627-2 (See Section 7.1)

Papers

V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950). English translation in: L. D. Landau, Collected papers (Oxford: Pergamon Press, 1965) p. 546
A.A. Abrikosov, Zh. Eksp. Teor. Fiz. 32, 1442 (1957) (English translation: Sov. Phys. JETP 5 1174 (1957)].) Abrikosov's original paper on vortex structure of Type-II superconductors derived as a solution of G–L equations for κ > 1/√2
L.P. Gor'kov, Sov. Phys. JETP 36, 1364 (1959)
A.A. Abrikosov's 2003 Nobel lecture: pdf file or video
V.L. Ginzburg's 2003 Nobel Lecture: pdf file or video
Gaiotto, Davide; Gukov, Sergei; Seiberg, Nathan (2013), "Surface Defects and Resolvents", Journal of High Energy Physics, arXiv:1307.2578

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