金兹堡朗道理论

Ginzburg Landau theory From Wikipedia the free encyclopedia In physics Ginzburg Landau theory named after Vitaly Lazarevich Ginzburg and Lev Landau is a mathematical physical theory used to describesuperconductivity 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 Contents hide 1Introduction 2Simple interpretation 3Coherence length and penetration depth 4Fluctuations in the Ginzburg Landau model 5Classification of superconductors based on Ginzburg Landau theory 6Landau Ginzburg theories in string theory 7See also 8References 8 1Papers Introduction edit 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 Fn is the free energy in the normal phase and 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 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 less electric current density and Re the real part The first equation which bears some similarities to the time independentSchr dinger equation but is principally different due to a nonlinear term determines the order parameter The second equation then provides the superconducting current Simple interpretation edit 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 state of the superconductor that is for temperatures above the superconducting transition temperature T Tc 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 T 0 T Tc with 0 0 Above the superconducting transition temperature T Tc 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 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 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 k 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 2 The exponential decay of the magnetic field is equivalent with the Higgs mechanism in high energy physics Fluctuations in the Ginzburg Landau model edit 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 edit 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 Hc1 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 Hc2 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 isquantized 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 citation needed Landau Ginzburg theories in string theory edit 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 onCalabi 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 Gaiotto Gukov Evgeny M Lifschitz 1984 Electrodynamics of Continuous Media Course of Theoretical Phy