In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after
Vitaly Ginzburg
Vitaly Lazarevich Ginzburg, ForMemRS (russian: Вита́лий Ла́заревич Ги́нзбург, link=no; 4 October 1916 – 8 November 2009) was a Russian physicist who was honored with the Nobel Prize in Physics in 2003, together wit ...
and
Lev Landau
Lev Davidovich Landau (russian: Лев Дави́дович Ланда́у; 22 January 1908 – 1 April 1968) was a Soviet-Azerbaijani physicist of Jewish descent who made fundamental contributions to many areas of theoretical physics.
His ac ...
, is a mathematical physical theory used to describe
superconductivity
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
. In its initial form, it was postulated as a phenomenological model which could describe
type-I superconductor
The interior of a bulk superconductor cannot be penetrated by a weak magnetic field, a phenomenon known as the Meissner effect. When the applied magnetic field becomes too large, superconductivity breaks down. Superconductors can be divided into ...
s without examining their microscopic properties. One GL-type superconductor is the famous
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 some limit of microscopic theory and giving microscopic interpretation of all its parameters. The theory can also be given a general geometric setting, placing it in the context of
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
, where in many cases exact solutions can be given. This general setting then extends to
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
and
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
, again owing to its solvability, and its close relation to other, similar systems.
Introduction
Based on
Landau
Landau ( pfl, Landach), officially Landau in der Pfalz, is an autonomous (''kreisfrei'') town surrounded by the Südliche Weinstraße ("Southern Wine Route") district of southern Rhineland-Palatinate, Germany. It is a university town (since 1990 ...
's previously established theory of second-order
phase transition
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states ...
s,
Ginzburg and Landau argued that the
free energy, ''F'', of a superconductor near the superconducting transition can be expressed in terms of a
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
order parameter field,
, where the quantity
is a measure of the local density, like a quantum mechanics
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
and
is nonzero below a phase transition into a superconducting state, although no direct interpretation of this parameter was given in the original paper. Assuming smallness of
and smallness of its
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
s, the
free energy has the form of a
field theory.
where ''F
n'' is the free energy in the normal phase, ''α'' and ''β'' in the initial argument were treated as phenomenological parameters,
is an
effective mass,
is an effective charge (usually 2''e'', where ''e'' is the charge of an electron),
is the
magnetic vector potential
In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic ...
, and
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
where j denotes the
dissipation
In thermodynamics, dissipation is the result of an irreversible process that takes place in homogeneous thermodynamic systems. In a dissipative process, energy ( internal, bulk flow kinetic, or system potential) transforms from an initial form to ...
-less
electric current density and ''Re'' the ''real part''. The first equation — which bears some similarities to the time-independent
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
, 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:
This equation has a trivial solution: . This corresponds to the normal conducting state, that is for temperatures above the superconducting transition temperature, .
Below the superconducting transition temperature, the above equation is expected to have a non-trivial solution (that is ). Under this assumption the equation above can be rearranged into:
When the right hand side of this equation is positive, there is a nonzero solution for (remember that the magnitude of a complex number can be positive or zero). This can be achieved by assuming the following temperature dependence of : with :
*Above the superconducting transition temperature, ''T'' > ''T''
''c'', the expression (''T'') / 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 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 . Furthermore,
that is approaches zero as ''T'' gets closer to ''T''
''c'' from below. Such a behavior is typical for a second order phase transition.
In Ginzburg–Landau theory the electrons that contribute to superconductivity were proposed to form a
superfluid
Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...
.
In this interpretation, , ,
2 indicates the fraction of electrons that have condensed into a superfluid.
[
]
Coherence length and penetration depth
The Ginzburg–Landau equations predicted two new characteristic lengths in a superconductor. The first characteristic length was termed coherence length
In physics, coherence length is the propagation distance over which a coherent wave (e.g. an electromagnetic wave) maintains a specified degree of coherence. Wave interference is strong when the paths taken by all of the interfering waves di ...
, ''ξ''. For ''T'' > ''Tc'' (normal phase), it is given by
:
while for ''T'' < ''Tc'' (superconducting phase), where it is more relevant, it is given by
:
It sets the exponential law according to which small perturbations of density of superconducting electrons recover their equilibrium value ''ψ''0. 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
The London equations, developed by brothers Fritz and Heinz London in 1935, are constitutive relations for a superconductor relating its superconducting current to electromagnetic fields in and around it. Whereas Ohm's law is the simplest const ...
. Expressed in terms of the parameters of Ginzburg–Landau model it is
:
where ''ψ''0 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 ''κ'' 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/, and Type II superconductor
In superconductivity, a type-II superconductor is a superconductor that exhibits an intermediate phase of mixed ordinary and superconducting properties at intermediate temperature and fields above the superconducting phases.
It also features the ...
s those with ''κ'' > 1/.
Fluctuations in the Ginzburg–Landau model
The phase transition
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states ...
from the normal state is of second order for Type II superconductors, taking into account fluctuations, 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
The Meissner effect (or Meissner–Ochsenfeld effect) is the expulsion of a magnetic field from a superconductor during its transition to the superconducting state when it is cooled below the critical temperature. This expulsion will repel a ne ...
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 ''Hc''. Depending on the geometry of the sample, one may obtain an intermediate state consisting of a baroque pattern of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. In Type II superconductor
In superconductivity, a type-II superconductor is a superconductor that exhibits an intermediate phase of mixed ordinary and superconducting properties at intermediate temperature and fields above the superconducting phases.
It also features the ...
s, raising the applied field past a critical value ''H''''c''1 leads to a mixed state (also known as the vortex state) in which an increasing amount of magnetic flux
In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted or . The SI unit of magnetic flux is the weber ...
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''''c''2, superconductivity is destroyed. The mixed state is actually caused by vortices in the electronic superfluid, sometimes called fluxon
In physics, a fluxon is a quantum of electromagnetic flux. The term may have any of several related meanings.
Superconductivity
In the context of superconductivity, in type II superconductors fluxons (also known as Abrikosov vortices) can for ...
s because the flux carried by these vortices is quantized. Most pure elemental superconductors, except niobium
Niobium is a chemical element with chemical symbol Nb (formerly columbium, Cb) and atomic number 41. It is a light grey, crystalline, and ductile transition metal. Pure niobium has a Mohs hardness rating similar to pure titanium, and it has s ...
and carbon nanotube
A scanning tunneling microscopy image of a single-walled carbon nanotube
Rotating single-walled zigzag carbon nanotube
A carbon nanotube (CNT) is a tube made of carbon with diameters typically measured in nanometers.
''Single-wall carbon na ...
s, 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.
Geometric formulation
The Ginzburg–Landau functional can be formulated in the general setting of a complex vector bundle over a compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
. This is the same functional as given above, transposed to the notation commonly used in Riemannian geometry. In multiple interesting cases, it can be shown to exhibit the same phenomena as the above, including Abrikosov vortices
In superconductivity, fluxon (also called a Abrikosov vortex and quantum vortex) is a vortex of supercurrent in a type-II superconductor, used by Alexei Abrikosov to explain magnetic behavior of type-II superconductors. Abrikosov vortices occu ...
(see discussion below).
For a complex vector bundle over a Riemannian manifold with fiber , the order parameter is understood as a section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
of the vector bundle . The Ginzburg–Landau functional is then a Lagrangian for that section:
:
The notation used here is as follows. The fibers are assumed to be equipped with a Hermitian inner product so that the square of the norm is written as . The phenomenological parameters and have been absorbed so that the potential energy term is a quartic mexican hat potential
Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the ...
; i.e., exhibiting spontaneous symmetry breaking
Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or ...
, with a minimum at some real value . The integral is explicitly over the volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of th ...
:
for an -dimensional manifold with determinant of the metric tensor .
The is the connection one-form and is the corresponding curvature 2-form (this is not the same as the free energy given up top; here, corresponds to the electromagnetic field strength tensor). The corresponds to the vector potential
In 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 gradient is a given vector field.
Formally, given a vector field v, a ''vecto ...
, but is in general non-Abelian when , and is normalized differently. In physics, one conventionally writes the connection as for the electric charge and vector potential ; in Riemannian geometry, it is more convenient to drop the (and all other physical units) and take to be a one-form
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to e ...
taking values in the Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
corresponding to the symmetry group of the fiber. Here, the symmetry group is SU(n)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the spec ...
, as that leaves the inner product invariant; so here, is a form taking values in the algebra .
The curvature generalizes the electromagnetic field strength to the non-Abelian setting, as the curvature form of an affine connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
on a vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
. It is conventionally written as
:
That is, each is an skew-symmetric matrix. (See the article on the metric connection
In mathematics, a metric connection is a connection in a vector bundle ''E'' equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported alo ...
for additional articulation of this specific notation.) To emphasize this, note that the first term of the Ginzburg–Landau functional, involving the field-strength only, is
:
which is just the Yang–Mills action on a compact Riemannian manifold.
The Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
s for the Ginzburg–Landau functional are the Yang–Mills equations
:
and
:
where is the Hodge star operator
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...
; i.e., the fully antisymmetric tensor. Note that these are closely related to the Yang–Mills–Higgs equations
In mathematics, the Yang–Mills–Higgs equations are a set of non-linear partial differential equations for a Yang–Mills field, given by a connection, and a Higgs field, given by a section of a vector bundle (specifically, the adjoint bundle) ...
.
Specific results
In string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
, it is conventional to study the Ginzburg–Landau functional for the manifold being a Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
, and taking ; i.e., a line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the '' tangent bundle'' is a way of organisi ...
. The phenomenon of Abrikosov vortices
In superconductivity, fluxon (also called a Abrikosov vortex and quantum vortex) is a vortex of supercurrent in a type-II superconductor, used by Alexei Abrikosov to explain magnetic behavior of type-II superconductors. Abrikosov vortices occu ...
persists in these general cases, including , where one can specify any finite set of points where vanishes, including multiplicity. The proof generalizes to arbitrary Riemann surfaces and to Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...
s. In the limit of weak coupling, it can be shown that converges uniformly
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
to 1, while and converge uniformly to zero, and the curvature becomes a sum over delta-function distributions at the vortices. The sum over vortices, with multiplicity, just equals the degree of the line bundle; as a result, one may write a line bundle on a Riemann surface as a flat bundle, with ''N'' singular points and a covariantly constant section.
When the manifold is four-dimensional, possessing a spin''c'' structure, then one may write a very similar functional, the Seiberg–Witten functional, which may be analyzed in a similar fashion, and which possesses many similar properties, including self-duality. When such systems are integrable
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
, they are studied as Hitchin system
In mathematics, the Hitchin integrable system is an integrable system depending on the choice of a complex reductive group and a compact Riemann surface, introduced by Nigel Hitchin in 1987. It lies on the crossroads of algebraic geometry, the ...
s.
Self-duality
When the manifold is a Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
, the functional can be re-written so as to explicitly show self-duality. One achieves this by writing the exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
as a sum of Dolbeault operator
In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients.
Complex forms have broad applications in differential geometry. On complex manifolds, ...
s . Likewise, the space of one-forms over a Riemann surface decomposes into a space that is holomorphic, and one that is anti-holomorphic: , so that forms in are holomorphic in and have no dependence on ; and ''vice-versa'' for . This allows the vector potential to be written as and likewise with and .
For the case of , where the fiber is so that the bundle is a line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the '' tangent bundle'' is a way of organisi ...
, the field strength can similarly be written as
:
Note that in the sign-convention being used here, both and are purely imaginary (''viz'' U(1)
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
is generated by so derivatives are purely imaginary). The functional then becomes
:
The integral is understood to be over the volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of th ...
:,
so that
:
is the total area of the surface . The is the Hodge star
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of ...
, as before. The degree of the line bundle over the surface is
:
where is the first Chern class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ...
.
The Lagrangian is minimized (stationary) when solve the Ginzberg–Landau equations
:
Note that these are both first-order differential equations, manifestly self-dual. Integrating the second of these, one quickly finds that a non-trivial solution must obey
:.
Roughly speaking, this can be interpreted as an upper limit to the density of the Abrikosov vortecies. One can also show that the solutions are bounded; one must have .
Landau–Ginzburg theories in string theory
In particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, any quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
with a unique classical vacuum state
In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
and a potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potenti ...
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
Cumrun Vafa ( fa, کامران وفا ; born 1 August 1960) is an Iranian-American theoretical physicist and the Hollis Professor of Mathematics and Natural Philosophy at Harvard University.
Early life and education
Cumrun Vafa was born in Tehran ...
and Nicholas Warner in November 1988; in this generalization one imposes that the superpotential
In theoretical physics, the superpotential is a function in supersymmetric quantum mechanics. Given a superpotential, two "partner potentials" are derived that can each serve as a potential in the Schrödinger equation. The partner potentials hav ...
possess a degenerate critical point. The same month, together with Brian Greene
Brian Randolph Greene (born February 9, 1963) is a American theoretical physicist, mathematician, and string theorist. Greene was a physics professor at Cornell University from 19901995, and has been a professor at Columbia University sinc ...
they argued that these theories are related by a renormalization group flow
In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in th ...
to sigma model
In physics, a sigma model is a field theory that describes the field as a point particle confined to move on a fixed manifold. This manifold can be taken to be any Riemannian manifold, although it is most commonly taken to be either a Lie group or ...
s on Calabi–Yau manifold
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstrin ...
s. In his 1993 paper "Phases of ''N'' = 2 theories in two-dimensions", Edward Witten
Edward Witten (born August 26, 1951) is an American mathematical and theoretical physicist. He is a Professor Emeritus in the School of Natural Sciences at the Institute for Advanced Study in Princeton. Witten is a researcher in string theory, q ...
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. 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.
See also
* Flux pinning
Flux pinning is a phenomenon that occurs when flux vortices in a type-II superconductor are prevented from moving within the bulk of the superconductor, so that the magnetic field lines are "pinned" to those locations. The superconductor must be a ...
* Gross–Pitaevskii equation The Gross–Pitaevskii equation (GPE, named after Eugene P. Gross and Lev Petrovich Pitaevskii) describes the ground state of a quantum system of identical bosons using the Hartree–Fock approximation and the pseudopotential interaction model.
...
* Landau theory
*Stuart–Landau equation The Stuart–Landau equation describes the behavior of a nonlinear oscillating system near the Hopf bifurcation, named after John Trevor Stuart and Lev Landau. In 1944, Landau proposed an equation for the evolution of the magnitude of the disturban ...
* Reaction–diffusion systems
* Quantum vortex
In physics, a quantum vortex represents a quantized flux circulation of some physical quantity. In most cases, quantum vortices are a type of topological defect exhibited in superfluids and superconductors. The existence of quantum vortices was ...
* Higgs bundle
In mathematics, a Higgs bundle is a pair (E,\varphi) consisting of a holomorphic vector bundle ''E'' and a Higgs field \varphi, a holomorphic 1-form taking values in the bundle of endomorphisms of ''E'' such that \varphi \wedge \varphi=0. Such pai ...
* Bogomol'nyi–Prasad–Sommerfield bound
References
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 superconductor
In superconductivity, a type-II superconductor is a superconductor that exhibits an intermediate phase of mixed ordinary and superconducting properties at intermediate temperature and fields above the superconducting phases.
It also features the ...
s 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
o
* V.L. Ginzburg's 2003 Nobel Lecture
pdf file
o
{{DEFAULTSORT:Ginzburg-Landau theory
Superconductivity
Quantum field theory
String theory
Lev Landau