Ginzburg-Landau theory
Encyclopedia
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.
s, 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
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
(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
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.
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 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:
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) = α0 (T - Tc) with α0 / β > 0:
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 superfluid
. In this interpretation |ψ|2 indicates the fraction of electrons that has condensed into a superfluid.
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 superconductor
s 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.
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.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, Ginzburg–Landau theory, named after Vitaly Lazarevich Ginzburg and Lev Landau
Lev Landau
Lev Davidovich Landau was a prominent Soviet physicist who made fundamental contributions to many areas of theoretical physics...
, is a mathematical theory used to model superconductivity
Superconductivity
Superconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...
. 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
Phenomenology (science)
The term phenomenology in science is used to describe a body of knowledge that relates empirical observations of phenomena to each other, in a way that 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 transitionPhase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
s, Landau
Lev Landau
Lev Davidovich Landau was a prominent Soviet physicist who made fundamental contributions to many areas of theoretical physics...
and Ginzburg
Vitaly Ginzburg
Vitaly Lazarevich Ginzburg ForMemRS was a Soviet theoretical physicist, astrophysicist, Nobel laureate, a member of the Russian Academy of Sciences and one of the fathers of Soviet hydrogen bomb...
argued that the free energy
Thermodynamic free energy
The thermodynamic free energy is the amount of work that a thermodynamic system can perform. The concept is useful in the thermodynamics of chemical or thermal processes in engineering and science. The free energy is the internal energy of a system less the amount of energy that cannot be used to...
F of a superconductor near the superconducting transition can be expressed in terms of a complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
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, e is the charge of an electron, A is the magnetic vector potential, and
(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 equationswhere j denotes the electrical current density and Re the real part. The first equation, which bears interesting similarities to the time-independent Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it 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 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:
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) = α0 (T - Tc) with α0 / β > 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
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 superfluid
Superfluid
Superfluidity is a state of matter in which the matter behaves like a fluid without viscosity and with extremely high thermal conductivity. The substance, which appears to be a normal liquid, will flow without friction past any surface, which allows it to continue to circulate over obstructions and...
. 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 lengthSuperconducting coherence length
A Superconducting coherence length, usually denoted as \xi, can be interpreted in two different ways:* It gives approximate spatial dimension of the Cooper pair,...
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 superconductor
Type I superconductor
Superconductors cannot be penetrated by magnetic flux lines . This 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...
s are those with 0 < κ < 1/√2, and Type II superconductors those with κ > 1/√2.
For Type II superconductors, the phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
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
Tricritical point
In condensed matter physics, dealing with the macroscopic physical properties of matter, a tricritical point is a point in the phase diagram of a system at whichthree-phase coexistence terminates...
).
The most important finding from Ginzburg–Landau theory was made by Alexei Abrikosov
Alexei Alexeyevich Abrikosov
Alexei Alexeyevich Abrikosov is a Soviet and 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 limit
Scaling limit
In 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. A lattice model which approximates a continuum quantum field theory in the limit as the lattice spacing goes to zero corresponds to finding a second...
of the XY model
XY model
The classical XY model is a model of statistical mechanics. It is the special case of the n-vector model for n=2.-Definition:...
.
The importance of the theory is also enhanced by a certain similarity with the Higgs mechanism
Higgs mechanism
In particle physics, the Higgs mechanism is the process in which gauge bosons in a gauge theory can acquire non-vanishing masses through absorption of Nambu-Goldstone bosons arising in spontaneous symmetry breaking....
in high-energy physics.
Landau–Ginzburg theories in particle physics
In particle physicsParticle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
any quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
with a unique classical vacuum state
Vacuum state
In quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...
and a potential energy
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...
with a degenerate critical point is called a Landau–Ginzburg theory. The generalization to N=(2,2) supersymmetric theories
Supersymmetry
In 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 Vafa
Cumrun Vafa
Cumrun Vafa 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 superpotential
Superpotential
Superpotential 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 Greene
Brian Greene
Brian Greene is an American theoretical physicist and string theorist. 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 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
Edward Witten
Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....
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). 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.Pierre-Gilles de GennesPierre-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 KleinertHagen KleinertHagen 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)