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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. The theory can also be given a general geometric setting, placing it in the context of Riemannian geometry, where in many cases exact solutions can be given. This general setting then extends to quantum field theory and string theory, again owing to its solvability, and its close relation to other, similar systems.

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.

where *Fn
• 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, andis 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-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:

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:

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 (TT**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,

that is ψ approaches zero as T gets closer to Tcfrom 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. The first characteristic length was termed coherence length, 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 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

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.

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 Hc. 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]

Geometric formulation

The Ginzburg–Landau functional can be formulated in the general setting of a complex vector bundle over a compact Riemannian manifold.[5] 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 (see discussion below).

For a complex vector bundleover a Riemannian manifoldwith fiber, the order parameteris understood as asectionof the vector bundle. The Ginzburg–Landau functional is then aLagrangianfor that section:
The notation used here is as follows. The fibersare assumed to be equipped with aHermitian inner productso that the square of the norm is written as. The phenomenological parametersandhave been absorbed so that the potential energy term is a quarticmexican hat potential, i.e. exhibitingspontaneous symmetry breaking, with a minimum at some real value. The integral is explicitly over thevolume form
for an-dimensional manifoldwith determinantof the metric tensor.
Theis theconnection one-formandis the correspondingcurvature 2-form(this is not the sameas given up top; here,corresponds to theelectromagneticfield strength tensor). Thecorresponds to thevector potential, but is in generalnon-Abelianwhen, and is normalized differently. In physics, one conventionally writes the connection asfor the electric chargeand vector potential; in Riemannian geometry, it is more convenient to drop the(and all other physical units) and taketo be aone-formtaking values in theLie algebracorresponding to the symmetry group of the fiber. Here, the symmetry group isSU(n), as that leaves the inner productinvariant; so here,is a form taking values in the algebra.
The curvaturegeneralizes theelectromagnetic field strengthto the non-Abelian setting, as thecurvature formof anaffine connectionon avector bundle. It is conventionally written as
That is, eachis anskew-symmetric matrix. (See the article on themetric connectionfor 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 equations for the Ginzburg–Landau functional are the Yang–Mills equations

and

whereis theHodge star operator, i.e. the fully antisymmetric tensor. Note that these are closely related to theYang–Mills–Higgs equations.

Specific results

Instring theory, it is conventional to study the Ginzburg–Landau functional for the manifoldbeing aRiemann surface, and taking, i.e. aline bundle.[6] The phenomenon ofAbrikosov vorticespersists in these general cases, including, where one can specify any finite set of points wherevanishes, including multiplicity.[7] The proof generalizes to arbitrary Riemann surfaces and toKähler manifolds.[8][9][10][11] In the limit of weak coupling, it can be shown thatconverges uniformlyto 1, whileandconverge uniformly to zero, and the curvature becomes a sum over delta-function distributions at the vortices.[12] 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 spinc 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, they are studied as Hitchin systems.

Self-duality

When the manifoldis aRiemann surface, the functional can be re-written so as to explicitly show self-duality. One achieves this by writing theexterior derivativeas a sum ofDolbeault operators. Likewise, the spaceof one-forms over a Riemann surface decomposes into a space that is holomorphic, and one that is anti-holomorphic:, so that forms inare holomorphic inand have no dependence on; and vice-versa for. This allows the vector potential to be written asand likewisewithand.
For the case of, where the fiber isso that the bundle is aline bundle, the field strength can similarly be written as
Note that in the sign-convention being used here, bothandare purely imaginary (vizU(1)is generated byso derivatives are purely imaginary). The functional then becomes

The integral is understood to be over the volume form

,

so that

is the total area of the surface. Theis theHodge star, as before. The degreeof the line bundleover the surfaceis
whereis the firstChern class.
The Lagrangian is minimized (stationary) whensolve 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 a 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, 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 [18] , 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 [19] . In his 1993 paper Phases of N = 2 theories in two-dimensions [20] , 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 [21] . 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.[13]

• Flux pinning

• Gross–Pitaevskii equation

• Landau theory

• Reaction–diffusion systems

• Quantum vortex

• Higgs bundle

References

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