In
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
,
Ginzburg–Landau theory is a mathematical theory used to model
superconductivitySuperconductivity is a phenomenon occurring in certain materials generally at very low temperatures, characterized by exactly zero electrical resistance and the exclusion of the interior magnetic field . It was discovered by Heike Kamerlingh Onnes in 1911. Like ferromagnetism and atomic spectral...
. 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
phenomenologicalThe term phenomenology in science is used to describe a body of knowledge which relates empirical observations of phenomena to each other, in a way which is consistent with fundamental theory, but is not directly derived from theory. For example, we find the following definition in the Concise...
as it describes some of the phenomena of superconductivity without explaining the underlying microscopic mechanism.
Based on Landau's previously-established theory of second-order
phase transitionA phase transition is a natural physical process. It has the characteristic of taking a given medium with given properties and transforming some or all of that medium, into a new medium with new properties. Phase transitions occur frequently and are found everywhere in the natural world...
s,
LandauLev Davidovich Landau was a prominent Soviet physicist who made fundamental contributions to many areas of theoretical physics...
and
GinzburgVitaly Lazarevich Ginzburg is a Russian theoretical physicist, astrophysicist and Nobel laureate and a member of the Russian Academy of Sciences...
argued that the
free energyIn thermodynamics, the term thermodynamic free energy refers to the amount of work that can be extracted from a system, and is helpful in engineering applications...
F of a superconductor near the superconducting transition can be expressed in terms of a
complexA complex number, in mathematics, is a number comprising a real number and an imaginary number; it can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit, having the property that i
2 = −1...
order parameter
ψ, which describes how deep into the superconducting phase the system is.
In
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
,
Ginzburg–Landau theory is a mathematical theory used to model
superconductivitySuperconductivity is a phenomenon occurring in certain materials generally at very low temperatures, characterized by exactly zero electrical resistance and the exclusion of the interior magnetic field . It was discovered by Heike Kamerlingh Onnes in 1911. Like ferromagnetism and atomic spectral...
. 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
phenomenologicalThe term phenomenology in science is used to describe a body of knowledge which relates empirical observations of phenomena to each other, in a way which is consistent with fundamental theory, but is not directly derived from theory. For example, we find the following definition in the Concise...
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 transitionA phase transition is a natural physical process. It has the characteristic of taking a given medium with given properties and transforming some or all of that medium, into a new medium with new properties. Phase transitions occur frequently and are found everywhere in the natural world...
s,
LandauLev Davidovich Landau was a prominent Soviet physicist who made fundamental contributions to many areas of theoretical physics...
and
GinzburgVitaly Lazarevich Ginzburg is a Russian theoretical physicist, astrophysicist and Nobel laureate and a member of the Russian Academy of Sciences...
argued that the
free energyIn thermodynamics, the term thermodynamic free energy refers to the amount of work that can be extracted from a system, and is helpful in engineering applications...
F of a superconductor near the superconducting transition can be expressed in terms of a
complexA complex number, in mathematics, is a number comprising a real number and an imaginary number; it can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit, having the property that i
2 = −1...
order parameter
ψ, which describes how deep into the superconducting phase the system is. The free energy has the form
where
Fn is the free energy in the normal phase,
α and
β are phenomenological parameters,
m is an effective mass,
A is the electromagnetic
vector potentialIn vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field....
, and (
B=curl(
A)) is the magnetic induction. By minimizing the free energy with respect to fluctuations in the order parameter and the vector potential, one arrives at the
Ginzburg–Landau equations
where
j denotes the electrical current density and
Re the
real part. The first equation, which bears interesting similarities to the time-independent
Schrödinger equationIn physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time...
, 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:
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
Tc.
Below 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:
.
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 α: α(
T) = α
0 (
T -
Tc) with α
0 / β > 0:
- Above 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 non-zero number, so only ψ = 0 solves the Ginzburg–Landau equation.
- Below 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
,
that is ψ approaches zero as
T gets closer to
Tc 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
superfluidSuperfluidity is a phase of matter or description of heat capacity in which unusual effects are observed when liquids, typically of helium-4 or helium-3, overcome friction by surface interaction when at a stage at which the liquid's viscosity becomes zero...
: In this interpretation |
ψ|
2 indicates the fraction of electrons that has condensed into a superfluid.
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 lengthIn physics, coherence length is the propagation distance from a coherent source to a point where an electromagnetic wave maintains a specified degree of coherence. The significance is that interference will be strong within a coherence length of the source, but not beyond it...
ξ, given by
which describes the size of thermodynamic fluctuations in the superconducting phase. The second is the
penetration depth λ, given by
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 superconductorType I superconductors are superconductors that cannot be penetrated by magnetic flux lines . As such, they have only a single critical temperature at which the material ceases to superconduct, becoming resistive. The origin of their superconductivity is fully explained by BCS theory...
s are those with 0 <
κ < 1/√2, and Type II superconductors those with
κ > 1/√2.
For Type II superconductors, the
phase transitionA phase transition is a natural physical process. It has the characteristic of taking a given medium with given properties and transforming some or all of that medium, into a new medium with new properties. Phase transitions occur frequently and are found everywhere in the natural world...
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 third textbook below).
The most important finding from Ginzburg–Landau theory was made by
Alexei AbrikosovAlexei Alexeyevich Abrikosov is a Russian theoretical physicist whose main contributions are in the field of condensed matter physics. He was awarded the Nobel Prize in Physics in 2003.-Biography :...
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 limitIn physics or mathematics, the scaling limit is a term applied to the behaviour of a lattice model in the limit of the lattice spacing going to zero....
of the
XY modelLike the famous Ising and Heisenberg models, the XY model is one of the many highly simplified models in statistical mechanics. It is a special case of the n-vector model. In the XY model, 2D classical spins are confined to some lattice. The spins are 2D unit vectors that obey O symmetry,...
.
The importance of the theory is also enhanced by a certain similarity with the
Higgs mechanismIn the standard model of particle physics, the Higgs mechanism is a theoretical framework which explains how the masses of the W and Z bosons arise as a result of electroweak symmetry breaking....
in high-energy physics.
Landau-Ginzburg theories in particle physics
In
particle physicsParticle physics is a branch of physics that studies the elementary constituents of matter and radiation, and the interactions between them. It is also called high energy physics, because many elementary particles do not occur under normal circumstances in nature, but can be created and detected...
any
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
with a unique classical
vacuum stateIn quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...
and a
potential energyPotential energy is energy stored within a physical system as a result of the position or configuration of the different parts of that system. It is called potential energy because it has the potential to be converted into other forms of energy, such as kinetic energy, and to do work in the process...
with a degenerate critical point is called a Landau-Ginzburg theory. The generalization to N=(2,2)
supersymmetric theoriesIn particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
in 2 spacetime dimensions was proposed by
Cumrun VafaCumrun Vafa کامران وفا is an Iranian-American leading string theorist from Harvard University where he started as a Harvard Junior Fellow. He is a recipient of the 2008 Dirac Medal.-Birth and education:...
and Nicholas Warner in the November 1988 article
Catastrophes and the Classification of Conformal Theories, in this generalization one imposes that the
superpotentialSuperpotential is a concept from particle physics' supersymmetry.- Example of superpotentiality:Let's look at the example of a one dimensional nonrelativistic particle with a 2D internal degree of freedom called "spin"...
possess a degenerate critical point. The same month, together with
Brian GreeneBrian Greene is an American theoretical physicist and one of the best-known string theorists. He has been a professor at Columbia University since 1996. Greene has worked on mirror symmetry, relating two different Calabi-Yau manifolds...
they argued that these theories are related by a renormalization group flow to
sigma modelIn physics, a sigma model is a physical system that is described by a Lagrangian density of the form:Depending on the scalars gij it is either a linear sigma model or a non-linear sigma model....
s on
Calabi-Yau manifoldIn mathematics and theoretical physics, Calabi–Yau manifolds or Calabi–Yau spaces are a certain important class of compact Kähler manifolds...
s in the paper
Calabi-Yau Manifolds and Renormalization Group Flows. In his 1993 paper
Phases of N=2 theories in two-dimensions,
Edward WittenEdward Witten is an American theoretical physicist and professor at the Institute for Advanced Study, who is widely known as “the most brilliant physicist of his generation”, and "one of the world's greatest living physicists, perhaps even Einstein's successor". He is a leading researcher in...
argued that Landau-Ginzburg theories and sigma models on Calabi-Yau manifolds are different phases of the same theory.
Papers
- V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950) ... Original paper of Ginzburg and Landau
- 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.
Pierre-Gilles de Gennes was a French physicist and the Nobel Prize laureate in Physics in 1991.-Biography:...
, 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
Hagen Kleinert is Professor of Theoretical Physics at the Free University of Berlin, Germany , at theWest University of Timişoara, at thein Bishkek. He is also of the...
, Gauge Fields in Condensed Matter, Vol. I World Scientific (Singapore, 1989); Paperback ISBN 9971-5-0210-0 (also available online here)