Ginzburg–Landau theory

In physics, Ginzburg–Landau theory, named after Vitaly Lazarevich Ginzburg and Lev Landau, is a mathematical theory used to model superconductivity. It does not purport to explain the microscopic mechanisms giving rise to superconductivity. Instead, it examines the macroscopic properties of a superconductor with the aid of general thermodynamic arguments.

This theory is sometimes called phenomenological as it describes some of the phenomena of superconductivity without explaining the underlying microscopic mechanism.

Introduction

Based on Landau's previously-established theory of second-order phase transitions, Landau and Ginzburg argued that the free energy F of a superconductor near the superconducting transition can be expressed in terms of a complex order parameter ψ, which describes how deep into the superconducting phase the system is. The free energy has the form

\( F = F_n + \alpha |\psi|^2 + \frac{\beta}{2} |\psi|^4 + \frac{1}{2m} \left| \left(-i\hbar\nabla - 2e\mathbf{A} \right) \psi \right|^2 + \frac{|\mathbf{B}|^2}{2\mu_0} \)

where Fn is the free energy in the normal phase, α and β are phenomenological parameters, m is an effective mass, e is the charge of an electron, A is the magnetic vector potential, and \( \mathbf{B}=\nabla \times \mathbf{A} (B=curl(A)) \) is the magnetic field. By minimizing the free energy with respect to fluctuations in the order parameter and the vector potential, one arrives at the Ginzburg–Landau equations

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

\( \mathbf{j} = \frac{2e}{m} \mathrm{Re} \left\{ \psi^* \left(-i\hbar\nabla - 2e \mathbf{A} \right) \psi \right\} \)

where j denotes the electrical current density and Re the real part. The first equation, which bears interesting similarities to the time-independent Schrödinger equation, determines the order parameter ψ based on the applied magnetic field. The second equation then provides the superconducting current.
Simple interpretation

Consider a homogeneous superconductor in absence of external magnetic field. Then 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 normal state of the superconductor, that is for temperatures T above the superconducting transition temperature \( T_{c} \).

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 non zero 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 \( \alpha : \alpha (T) = \alpha_{0} (T - T_{c}) \) with \( \alpha_{0} / \beta \)> 0:

Above the superconducting transition temperature, T > Tc, 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 < Tc, 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 Tc from below. Such a behaviour is typical for a second order phase transition.

I

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 has condensed into a superfluid.^[1]

Coherence length and penetration depth

The Ginzburg–Landau equations produce many interesting and valid results. Perhaps the most important of these is its prediction of the existence of two characteristic lengths in a superconductor. The first is a coherence length ξ, given by

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

which describes the size of thermodynamic fluctuations in the superconducting phase. The second is the penetration depth λ, given by

\( \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 describes the depth to which an external magnetic field can penetrate the superconductor.

The ratio κ = λ/ξ is known as the Ginzburg–Landau parameter. It has been shown that Type I superconductors are those with 0 < κ < 1/√2, and Type II superconductors those with κ > 1/√2.

For Type II superconductors, the phase transition from the normal state is of second order, for Type I superconductors it is of first order. This is proved by deriving a dual Ginzburg–Landau theory for the superconductor (see Chapter 13 of the last textbook below, and the Wikipedia entry Tricritical point).

The most important finding from Ginzburg–Landau theory was made by Alexei Abrikosov in 1957. In a type-II superconductor in a high magnetic field – the field penetrates in quantized tubes of flux, which are most commonly arranged in a hexagonal arrangement.

This theory arises as the scaling limit of the XY model. The importance of the theory is also enhanced by a certain similarity with the Higgs mechanism in high-energy physics.
See also

Gross–Pitaevskii equation
Landau theory
Reaction–diffusion systems

Landau–Ginzburg theories in particle physics

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.
References

^ a b 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. doi:10.1002/cphc.200400182. PMID 15298379.

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

Books

D. Saint-James, G. Sarma and E. J. Thomas, Type II Superconductivity Pergamon (Oxford 1969)
M. Tinkham, Introduction to Superconductivity, McGraw–Hill (New York 1996)
de Gennes, P.G., Superconductivity of Metals and Alloys, Perseus Books, 2nd Revised Edition (1995), ISBN 0-201-40842-2 (this book is heavily based on G–L theory)
Hagen Kleinert, Gauge Fields in Condensed Matter, Vol. I World Scientific (Singapore, 1989); Paperback ISBN 9971-5-0210-0 (also available online here)

Physics Encyclopedia