Ginzburg-Landau Theory

Get Ginzburg%E2%80%93Landau Theory essential facts below. View Videos or join the Ginzburg%E2%80%93Landau Theory discussion. Add Ginzburg%E2%80%93Landau Theory to your PopFlock.com topic list for future reference or share this resource on social media.
## Introduction

## Simple interpretation

## Coherence length and penetration depth

## Fluctuations in the Ginzburg-Landau model

## Classification of superconductors based on Ginzburg-Landau theory

## Geometric formulation

### Specific results

## Self-duality

## Landau-Ginzburg theories in string theory

## See also

## References

### Papers

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Ginzburg%E2%80%93Landau 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. One GL-type superconductor is the famous YBCO, and generally all Cuprates.^{[1]}

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.

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, *?*, 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 |*?*| and smallness of its gradients, the free energy has the form of a field theory.

where *F _{n}* is the free energy in the normal phase,

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, *?*. The second equation then provides the superconducting current.

Consider a homogeneous superconductor where there is no superconducting current and the equation for *?* simplifies to:

This equation has a trivial solution: *?* = 0. This corresponds to the normal conducting state, 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 *?* ? 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 *?* (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 *?*_{0}/*?* > 0:

- 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*?*= 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*?*. Furthermore,

- that is
*?*approaches zero as*T*gets closer to*T*_{c}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.^{[2]} In this interpretation, |*?*|^{2} indicates the fraction of electrons that have condensed into a superfluid.^{[2]}

The Ginzburg-Landau equations predicted two new characteristic lengths in a superconductor. The first characteristic length was termed **coherence length**, *?*. For *T* > *T _{c}* (normal phase), it is given by

while for *T* < *T _{c}* (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 *?*_{0}. Thus this theory characterized all superconductors by two length scales. The second one is the **penetration depth**, *?*. 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 *?*_{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 *?* belongs to Landau. The ratio *?* = *?*/*?* is presently known as the **Ginzburg-Landau parameter**. It has been proposed by Landau that Type I superconductors are those with 0 < *?* < 1/, and Type II superconductors those with *?* > 1/.

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.

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 *H _{c}*. Depending on the geometry of the sample, one may obtain an intermediate state

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.^{[5]}

The Ginzburg-Landau functional can be formulated in the general setting of a complex vector bundle over a compact Riemannian manifold.^{[6]} 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 bundle over a Riemannian manifold with fiber , the order parameter is understood as a section 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, *i.e.* exhibiting spontaneous symmetry breaking, with a minimum at some real value . The integral is explicitly over the volume form

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, 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 taking values in the Lie algebra corresponding to the symmetry group of the fiber. Here, the symmetry group is SU(n), 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 on a vector bundle . It is conventionally written as

That is, each is an skew-symmetric matrix. (See the article on the metric connection 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 equations for the Ginzburg-Landau functional are the Yang-Mills equations

and

where is the Hodge star operator, i.e. the fully antisymmetric tensor. Note that these are closely related to the Yang-Mills-Higgs equations.

In string theory, it is conventional to study the Ginzburg-Landau functional for the manifold being a Riemann surface, and taking , *i.e.* a line bundle.^{[7]} The phenomenon of Abrikosov vortices persists in these general cases, including , where one can specify any finite set of points where vanishes, including multiplicity.^{[8]} The proof generalizes to arbitrary Riemann surfaces and to Kähler manifolds.^{[9]}^{[10]}^{[11]}^{[12]} In the limit of weak coupling, it can be shown that converges uniformly to 1, while and converge uniformly to zero, and the curvature becomes a sum over delta-function distributions at the vortices.^{[13]} 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, they are studied as Hitchin systems.

When the manifold is a Riemann surface , the functional can be re-written so as to explicitly show self-duality. One achieves this by writing the exterior derivative as a sum of Dolbeault operators . 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, 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) is generated by so 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 . The is the Hodge star, as before. The degree of the line bundle over the surface is

where is the first Chern class.

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 .

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. 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. 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.^{[14]}

**^**Wesche, Chapter 50: High Temperature Superconductors, Springer 2017, at p. 1233, contained in Casap, Kapper Handbook- ^
^{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. **^**Lev D. Landau; Evgeny M. Lifschitz (1984).*Electrodynamics of Continuous Media*. Course of Theoretical Physics.**8**. Oxford: Butterworth-Heinemann. ISBN 978-0-7506-2634-7.**^**David J. E. Callaway (1990). "On the remarkable structure of the superconducting intermediate state".*Nuclear Physics B*.**344**(3): 627-645. Bibcode:1990NuPhB.344..627C. doi:10.1016/0550-3213(90)90672-Z.**^**Abrikosov, A. A. (1957). The magnetic properties of superconducting alloys. Journal of Physics and Chemistry of Solids, 2(3), 199-208.**^**Jost, Jürgen (2002). "The Ginzburg-Landau Functional".*Riemannian Geometry and Geometric Analysis*(Third ed.). Springer-Verlag. pp. 373-381. ISBN 3-540-42627-2.**^**Hitchin, N. J. (1987). "The Self-Duality Equations on a Riemann Surface".*Proceedings of the London Mathematical Society*. s3-55 (1): 59-126. doi:10.1112/plms/s3-55.1.59. ISSN 0024-6115.**^**Taubes, Clifford Henry (1980). "Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations".*Communications in Mathematical Physics*. Springer Science and Business Media LLC.**72**(3): 277-292. doi:10.1007/bf01197552. ISSN 0010-3616. S2CID 122086974.**^**Bradlow, Steven B. (1990). "Vortices in holomorphic line bundles over closed Kähler manifolds".*Communications in Mathematical Physics*. Springer Science and Business Media LLC.**135**(1): 1-17. doi:10.1007/bf02097654. ISSN 0010-3616. S2CID 59456762.**^**Bradlow, Steven B. (1991). "Special metrics and stability for holomorphic bundles with global sections".*Journal of Differential Geometry*. International Press of Boston.**33**(1): 169-213. doi:10.4310/jdg/1214446034. ISSN 0022-040X.**^**García-Prada, Oscar (1993). "Invariant connections and vortices".*Communications in Mathematical Physics*. Springer Science and Business Media LLC.**156**(3): 527-546. doi:10.1007/bf02096862. ISSN 0010-3616. S2CID 122906366.**^**García-Prada, Oscar (1994). "A Direct Existence Proof for the Vortex Equations Over a Compact Riemann Surface".*Bulletin of the London Mathematical Society*. Wiley.**26**(1): 88-96. doi:10.1112/blms/26.1.88. ISSN 0024-6093.**^**M.C. Hong, J, Jost, M Struwe, "Asymptotic limits of a Ginzberg-Landau type functional",*Geometric Analysis and the Calculus of Variations for Stefan Hildebrandt*(1996) International press (Boston) pp. 99-123.**^**Gaiotto, Davide; Gukov, Sergei; Seiberg, Nathan (2013), "Surface Defects and Resolvents",*Journal of High Energy Physics*,**2013**(9): 70, arXiv:1307.2578, Bibcode:2013JHEP...09..070G, doi:10.1007/JHEP09(2013)070, S2CID 118498045

- 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 ? > 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

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Popular Products

Music Scenes

Popular Artists