Ginzburg–Landau theory
In physics, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after Vitaly 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. 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, formula_1, where the quantity formula_2 is  a measure of the local density, like a quantum mechanics wave function and formula_3 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 formula_4 and smallness of its gradients, the free energy has the form of a field theory. while for "T" < "T" (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". 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 "ψ" 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 superconductors those with "κ" > 1/. Fluctuations in the Ginzburg–Landau model. The phase 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. 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 "H". 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 superconductors, raising the applied field past a critical value "H" 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", 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. Geometric formulation. The Ginzburg–Landau functional can be formulated in the general setting of a complex vector bundle over a compact Riemannian manifold. 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 formula_8 over a Riemannian manifold formula_9 with fiber formula_10, the order parameter formula_11 is understood as a section of the vector bundle formula_8. The Ginzburg–Landau functional is then a Lagrangian for that section: The notation used here is as follows. The fibers formula_10 are assumed to be equipped with a Hermitian inner product formula_15 so that the square of the norm is written as formula_16. The phenomenological parameters formula_17 and formula_18 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 formula_19. The integral is explicitly over the volume form for an formula_21-dimensional manifold formula_9 with determinant formula_23 of the metric tensor formula_24. The formula_25 is the connection one-form and formula_26 is the corresponding curvature 2-form (this is not the same as the free energy formula_26 given up top; here, formula_26 corresponds to the electromagnetic field strength tensor). The formula_29 corresponds to the vector potential, but is in general non-Abelian when formula_30, and is normalized differently. In physics, one  conventionally writes the connection as formula_31 for the electric charge formula_32 and vector potential formula_29; in Riemannian geometry, it is more convenient to drop the formula_32 (and all other physical units) and take formula_35 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 formula_15 invariant; so here, formula_29 is a form taking values in the algebra formula_38. The curvature formula_26 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 formula_41 is an formula_42 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 formula_46 is the Hodge star operator; i.e., the fully antisymmetric tensor. Note that these are closely related to the Yang –Mills–Higgs equations. Specific results. In string theory, it is conventional to study the Ginzburg–Landau functional for the manifold formula_9 being a Riemann surface, and taking formula_48; i.e., a line bundle. The phenomenon of Abrikosov vortices persists in these general cases, including formula_49, where one can specify any finite set of points where formula_11 vanishes, including multiplicity. The proof generalizes to arbitrary Riemann surfaces and to Kähler manifolds. In the limit of weak coupling, it can be shown that formula_51 converges uniformly to 1, while formula_52 and formula_53 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 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  manifold formula_9 is a Riemann surface formula_55, 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 formula_56. Likewise, the space formula_57 of one-forms over a Riemann surface decomposes into a space that is holomorphic, and one that is anti-holomorphic: formula_58, so that forms in formula_59 are holomorphic in formula_60 and have no dependence on formula_61; and "vice-versa" for formula_62. This allows the vector potential to be written as formula_63 and likewise formula_64 with formula_65 and formula_66. For the case  of formula_67, where the fiber is formula_68 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 formula_70 and formula_26 are purely imaginary ("viz" U(1) is generated by formula_72 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 formula_76. The formula_46 is the Hodge star, as before. The degree formula_78 of the line bundle formula_79 over the surface formula_76 is where formula_82 is the first  Chern class. The Lagrangian is minimized (stationary) when formula_83 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 formula_86. 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 November 1988; 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 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. 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.