Inphysics,Ginzburg–Landau theory, often calledLandau–Ginzburg theory, named afterVitaly Ginzburg andLev Landau, is a mathematical physical theory used to describesuperconductivity. In its initial form, it was postulated as a phenomenological model which could describetype-I superconductors without examining their microscopic properties. One GL-type superconductor is the famousYBCO, and generally allcuprates.^[1]

Later, a version of Ginzburg–Landau theory was derived from theBardeen–Cooper–Schrieffer microscopic theory byLev Gor'kov,^[2] 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 ofRiemannian geometry, where in many cases exact solutions can be given. This general setting then extends toquantum field theory andstring theory, again owing to its solvability, and its close relation to other, similar systems.

Based onLandau'spreviously established theory of second-orderphase transitions,Ginzburg and Landau argued that thefree energy density${\displaystyle f_s}$ of a superconductor near the superconducting transition can be expressed in terms of acomplexorder parameter field${\displaystyle \psi (r)=|\psi (r)|e^{i\varphi (r)}}$, where the quantity${\displaystyle |\psi (r){|}^2}$ is a measure of the local density of superconducting electrons${\displaystyle n_s(r)}$ analogous to a quantum mechanicalwave function.^[2] While${\displaystyle \psi (r)}$ is nonzero below a phase transition into a superconducting state, no direct interpretation of this parameter was given in the original paper. Assuming smallness of${\displaystyle |\psi |}$ and smallness of itsgradients, thefree energy density has the form of afield theory and exhibits U(1) gauge symmetry:

$${\displaystyle f_s=f_n+\alpha (T)|\psi {|}^2+\frac{1}{2}\beta (T)|\psi {|}^4+\frac{1}{2m^{\ast }}{\left|\left(-i\hslash \mathrm{\nabla }-\frac{e^{\ast }}{c}\mathbf{A}\right)\psi \right|}^2+\frac{{\mathbf{B}}^2}{8\pi },}$$

where

• ${\displaystyle f_n}$ is the free energy density of the normal phase,
• ${\displaystyle \alpha (T)}$ and${\displaystyle \beta (T)}$ are phenomenological parameters that are functions of T (and often written just${\displaystyle \alpha }$ and${\displaystyle \beta }$).
• ${\displaystyle m^{\ast }}$ is aneffective mass,
• ${\displaystyle e^{\ast }}$ is an effective charge (usually${\displaystyle 2e}$, where e is the charge of an electron),
• ${\displaystyle \mathbf{A}}$ is themagnetic vector potential, and
• ${\displaystyle \mathbf{B}=\mathrm{\nabla }\times \mathbf{A}}$ is the magnetic field.

The total free energy is given by${\displaystyle F=\int f_s{d}^{3}r}$. By minimizing${\displaystyle F}$ with respect to variations in the order parameter${\displaystyle \psi }$ and the vector potential${\displaystyle \mathbf{A}}$, one arrives at theGinzburg–Landau equations:

$${\displaystyle \alpha \psi +\beta |\psi {|}^2\psi +\frac{1}{2m^{\ast }}{\left(-i\hslash \mathrm{\nabla }-\frac{e^{\ast }}{c}\mathbf{A}\right)}^2\psi =0}$$

$${\displaystyle \mathrm{\nabla }\times \mathbf{B}=\frac{4\pi }{c}\mathbf{J};\mathbf{J}=\frac{e^{\ast }}{m^{\ast }}\mathrm{Re}\left\{{\psi }^{\ast }\left(-i\hslash \mathrm{\nabla }-\frac{e^{\ast }}{c}\mathbf{A}\right)\psi \right\},}$$

where${\displaystyle J}$ denotes thedissipation-freeelectric current density andRe thereal part. The first equation — which bears some similarities to the time-independentSchrödinger equation, but is principally different due to a nonlinear term — determines the order parameter,${\displaystyle \psi }$. The second equation then provides the superconducting current.

Consider a homogeneous superconductor where there is no superconducting current and the equation forψ simplifies to:$${\displaystyle \alpha \psi +\beta |\psi {|}^2\psi =0.}$$

This equation has a trivial solution:ψ = 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${\displaystyle \psi \ne 0}$). Under this assumption the equation above can be rearranged into:$${\displaystyle |\psi {|}^2=-\frac{\alpha }{\beta }.}$$

When the right hand side of this equation is positive, there is a nonzero solution forψ (remember that the magnitude of a complex number can be positive or zero). This can be achieved by assuming the following temperature dependence of${\displaystyle \alpha :\alpha (T)={\alpha }_0(T-T_{\mathrm{c}})}$with${\displaystyle {\alpha }_0/\beta >0}$:

• Above the superconducting transition temperature,T >T_c, the expressionα(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 onlyψ = 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 forψ. Furthermore,$${\displaystyle |\psi {|}^2=-\frac{{\alpha }_0(T-T_c)}{\beta },}$$ that isψ approaches zero asT gets closer toT_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 asuperfluid.^[3] In this interpretation, |ψ|² indicates the fraction of electrons that have condensed into a superfluid.^[3]

The Ginzburg–Landau equations predicted two new characteristic lengths in a superconductor. The first characteristic length was termedcoherence length,ξ. ForT >T_c (normal phase), it is given by

${\displaystyle \xi =\sqrt{\frac{{\hslash }^2}{2m^{\ast }|\alpha |}}.}$

while forT <T_c (superconducting phase), where it is more relevant, it is given by

${\displaystyle \xi =\sqrt{\frac{{\hslash }^2}{4m^{\ast }|\alpha |}}.}$

It sets the exponential law according to which small perturbations of density of superconducting electrons recover their equilibrium valueψ₀. 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 theirLondon theory. Expressed in terms of the parameters of Ginzburg–Landau model it is

${\displaystyle \lambda =\sqrt{\frac{m^{\ast }}{{\mu }_0{e}^{\ast 2}{\psi }_0^2}}=\sqrt{\frac{m^{\ast }\beta }{{\mu }_0{e}^{\ast 2}|\alpha |}},}$

whereψ₀ 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 parameterκ belongs to Landau. The ratioκ =λ/ξ is presently known as the Ginzburg–Landau parameter. It has been proposed by Landau thatType I superconductors are those with 0 <κ < 1/√2, andType II superconductors those withκ > 1/√2.

Thephase transition 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.^[4]

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. TheMeissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. InType I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical valueH_c. Depending on the geometry of the sample, one may obtain an intermediate state^[5] consisting of a pattern of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. InType II superconductors, raising the applied field past a critical valueH_c1 leads to a mixed state (also known as the vortex state) in which an increasing amount ofmagnetic 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 strengthH_c2, superconductivity is destroyed. The mixed state is actually caused by vortices in the electronic superfluid, sometimes calledfluxons because the flux carried by these vortices isquantized. Most pureelemental superconductors, exceptniobium andcarbon nanotubes, are Type I, while almost all impure and compound superconductors are Type II.

The most important finding from Ginzburg–Landau theory was made byAlexei 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 fluxvortices.^[6]

The Ginzburg–Landau functional can be formulated in the general setting of acomplex vector bundle over acompactRiemannian manifold.^[7] 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, includingAbrikosov vortices (see discussion below).

For a complex vector bundle${\displaystyle E}$ over a Riemannian manifold${\displaystyle M}$ with fiber${\displaystyle {\mathbb{C}}^n}$, the order parameter${\displaystyle \psi }$ is understood as asection of the vector bundle${\displaystyle E}$. The Ginzburg–Landau functional is then aLagrangian for that section:

${\displaystyle \mathcal{L}(\psi ,A)={\int }_M\sqrt{|g|}d{x}^1\wedge \cdots \wedge d{x}^m\left[|F{|}^2+|D\psi {|}^2+\frac{1}{4}{\left(\sigma -|\psi {|}^2\right)}^2\right]}$

The notation used here is as follows. The fibers${\displaystyle {\mathbb{C}}^n}$ are assumed to be equipped with aHermitian inner product${\displaystyle ⟨\cdot ,\cdot ⟩}$ so that the square of the norm is written as${\displaystyle |\psi {|}^2=⟨\psi ,\psi ⟩}$. The phenomenological parameters${\displaystyle \alpha }$ and${\displaystyle \beta }$ have 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${\displaystyle \sigma \in \mathbb{R}}$. The integral is explicitly over thevolume form

${\displaystyle \ast (1)=\sqrt{|g|}d{x}^1\wedge \cdots \wedge d{x}^m}$

for an${\displaystyle m}$-dimensional manifold${\displaystyle M}$ with determinant${\displaystyle |g|}$ of the metric tensor${\displaystyle g}$.

The${\displaystyle D=d+A}$ is theconnection one-form and${\displaystyle F}$ is the correspondingcurvature 2-form (this is not the same as the free energy${\displaystyle F}$ given up top; here,${\displaystyle F}$ corresponds to theelectromagneticfield strength tensor). The${\displaystyle A}$ corresponds to thevector potential, but is in generalnon-Abelian when${\displaystyle n>1}$, and is normalized differently. In physics, one conventionally writes the connection as${\displaystyle d-ieA}$ for the electric charge${\displaystyle e}$ and vector potential${\displaystyle A}$; in Riemannian geometry, it is more convenient to drop the${\displaystyle e}$ (and all other physical units) and take${\displaystyle A=A_{\mu }d{x}^{\mu }}$ to be aone-form taking values in theLie algebra corresponding to the symmetry group of the fiber. Here, the symmetry group isSU(n), as that leaves the inner product${\displaystyle ⟨\cdot ,\cdot ⟩}$ invariant; so here,${\displaystyle A}$ is a form taking values in the algebra${\displaystyle \mathfrak{s}\mathfrak{u}(n)}$.

The curvature${\displaystyle F}$ generalizes theelectromagnetic field strength to the non-Abelian setting, as thecurvature form of anaffine connection on avector bundle . It is conventionally written as

${\displaystyle \begin{array}{r}F=D\circ D=dA+A\wedge A=\left(\frac{\mathrm{\partial }{A}_{\nu }}{\mathrm{\partial }{x}^{\mu }}+A_{\mu }{A}_{\nu }\right)d{x}^{\mu }\wedge d{x}^{\nu }=\frac{1}{2}\left(\frac{\mathrm{\partial }{A}_{\nu }}{\mathrm{\partial }{x}^{\mu }}-\frac{\mathrm{\partial }{A}_{\mu }}{\mathrm{\partial }{x}^{\nu }}+[A_{\mu },A_{\nu }]\right)d{x}^{\mu }\wedge d{x}^{\nu }\end{array}}$

That is, each${\displaystyle A_{\mu }}$ is an${\displaystyle n\times n}$ skew-symmetric matrix. (See the article on themetric 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

${\displaystyle \mathcal{L}(A)=YM(A)={\int }_M\ast (1)|F{|}^2}$

which is just theYang–Mills action on a compact Riemannian manifold.

TheEuler–Lagrange equations for the Ginzburg–Landau functional are the Yang–Mills equations^[8]

${\displaystyle D^{\ast }D\psi =\frac{1}{2}\left(\sigma -|\psi {|}^2\right)\psi }$

and

${\displaystyle D^{\ast }F=-\mathrm{Re}⟨D\psi ,\psi ⟩}$

where${\displaystyle D^{\ast }}$ is theadjoint of${\displaystyle D}$, analogous to thecodifferential${\displaystyle \delta =d^{\ast }}$. Note that these are closely related to theYang–Mills–Higgs equations.

Instring theory, it is conventional to study the Ginzburg–Landau functional for the manifold${\displaystyle M}$ being aRiemann surface, and taking${\displaystyle n=1}$; i.e., aline bundle.^[9] The phenomenon ofAbrikosov vortices persists in these general cases, including${\displaystyle M={\mathbb{R}}^2}$, where one can specify any finite set of points where${\displaystyle \psi }$ vanishes, including multiplicity.^[10] The proof generalizes to arbitrary Riemann surfaces and toKähler manifolds.^{[11][12][13][14]} In the limit of weak coupling, it can be shown that${\displaystyle |\psi |}$converges uniformly to 1, while${\displaystyle D\psi }$ and${\displaystyle dA}$ converge uniformly to zero, and the curvature becomes a sum over delta-function distributions at the vortices.^[15] 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, withN singular points and a covariantly constant section.

When the manifold is four-dimensional, possessing aspin^c structure, then one may write a very similar functional, theSeiberg–Witten functional, which may be analyzed in a similar fashion, and which possesses many similar properties, including self-duality. When such systems areintegrable, they are studied asHitchin systems.

When the manifold${\displaystyle M}$ is aRiemann surface${\displaystyle M=\mathrm{\Sigma }}$, the functional can be re-written so as to explicitly show self-duality. One achieves this by writing theexterior derivative as a sum ofDolbeault operators${\displaystyle d=\mathrm{\partial }+\overline{\mathrm{\partial }}}$. Likewise, the space${\displaystyle {\mathrm{\Omega }}^1}$ of one-forms over a Riemann surface decomposes into a space that is holomorphic, and one that is anti-holomorphic:${\displaystyle {\mathrm{\Omega }}^1={\mathrm{\Omega }}^{1,0}\oplus {\mathrm{\Omega }}^{0,1}}$, so that forms in${\displaystyle {\mathrm{\Omega }}^{1,0}}$ are holomorphic in${\displaystyle z}$ and have no dependence on${\displaystyle \overline{z}}$; andvice-versa for${\displaystyle {\mathrm{\Omega }}^{0,1}}$. This allows the vector potential to be written as${\displaystyle A=A^{1,0}+A^{0,1}}$ and likewise${\displaystyle D={\mathrm{\partial }}_A+{\overline{\mathrm{\partial }}}_A}$ with${\displaystyle {\mathrm{\partial }}_A=\mathrm{\partial }+A^{1,0}}$ and${\displaystyle {\overline{\mathrm{\partial }}}_A=\overline{\mathrm{\partial }}+A^{0,1}}$.

For the case of${\displaystyle n=1}$, where the fiber is${\displaystyle \mathbb{C}}$ so that the bundle is aline bundle, the field strength can similarly be written as

${\displaystyle F=-\left({\mathrm{\partial }}_A{\overline{\mathrm{\partial }}}_A+{\overline{\mathrm{\partial }}}_A{\mathrm{\partial }}_A\right)}$

Note that in the sign-convention being used here, both${\displaystyle A^{1,0},A^{0,1}}$ and${\displaystyle F}$ are purely imaginary (vizU(1) is generated by${\displaystyle e^{i\theta }}$ so derivatives are purely imaginary). The functional then becomes

${\displaystyle \mathcal{L}\left(\psi ,A\right)=2\pi \sigma \mathrm{deg}L+{\int }_{\mathrm{\Sigma }}\frac{i}{2}dz\wedge d\overline{z}\left[2|{\overline{\mathrm{\partial }}}_A\psi {|}^2+{\left(\ast (-iF)-\frac{1}{2}(\sigma -|\psi {|}^2\right)}^2\right]}$

The integral is understood to be over thevolume form

${\displaystyle \ast (1)=\frac{i}{2}dz\wedge d\overline{z}}$,

so that

${\displaystyle \mathrm{Area}\mathrm{\Sigma }={\int }_{\mathrm{\Sigma }}\ast (1)}$

is the total area of the surface${\displaystyle \mathrm{\Sigma }}$. The${\displaystyle \ast }$ is theHodge star, as before. The degree${\displaystyle \mathrm{deg}L}$ of the line bundle${\displaystyle L}$ over the surface${\displaystyle \mathrm{\Sigma }}$ is

${\displaystyle \mathrm{deg}L=c_1(L)=\frac{1}{2\pi }{\int }_{\mathrm{\Sigma }}iF}$

where${\displaystyle c_1(L)=c_1(L)[\mathrm{\Sigma }]\in H^2(\mathrm{\Sigma })}$ is the firstChern class.

The Lagrangian is minimized (stationary) when${\displaystyle \psi ,A}$ 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

${\displaystyle 4\pi \mathrm{deg}L\le \sigma \mathrm{Area}\mathrm{\Sigma }}$.

Roughly speaking, this can be interpreted as an upper limit to the density of the Abrikosov vortices. One can also show that the solutions are bounded; one must have${\displaystyle |\psi |\le \sigma }$.

Inparticle physics, anyquantum field theory with a unique classicalvacuum state and apotential energy with adegenerate critical point is called a Landau–Ginzburg theory. The generalization toN = (2,2)supersymmetric theories in 2 spacetime dimensions was proposed byCumrun Vafa andNicholas Warner in November 1988;^[16] in this generalization one imposes that thesuperpotential possess a degenerate critical point. The same month, together withBrian Greene they argued that these theories are related by arenormalization group flow tosigma models onCalabi–Yau manifolds.^[17] In his 1993 paper "Phases ofN = 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.^[18] 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.^[19] 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.^[20]

• 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. JETP5 1174 (1957)].) Abrikosov's original paper on vortex structure ofType-II superconductors derived as a solution of G–L equations for κ > 1/√2
• L.P. Gor'kov,Sov. Phys. JETP36, 1364 (1959)
• A.A. Abrikosov's 2003 Nobel lecture:pdf file orvideo
• V.L. Ginzburg's 2003 Nobel Lecture:pdf file orvideo

• Superconductivity
• Quantum field theory
• Lev Landau

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