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.
== 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.
F =
F
n
+ α
|
ψ
|
2
+
β 2
|
ψ
|
4
+
1
2 m
|
(
− i
ℏ ∇
− 2
e
A
)
ψ
|
2
+
|
B
|
2
μ
0
{\displaystyle F=F_{n}+\alpha |\psi |^{2}+{\frac {2}}|\psi |^{4}+{\frac {1}{2m}}\left|\left(-i\hbar
\nabla -2e\mathbf {A} \right)\psi \right|^{2}+{\frac {|\mathbf {B} |^{2}}{2\mu _{0}}}}
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, and
B
= ∇
×
A
{\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }
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
α ψ
+ β
|
ψ
|
2
ψ +
1
2 m
(
− i
ℏ ∇
− 2
e
A
)
2
ψ =
0
{\displaystyle \alpha \psi +\beta |\psi |^{2}\psi +{\frac {1}{2m}}\left(-i\hbar \nabla -2e\mathbf
{A} \right)^{2}\psi =0}
∇ ×
B
=
μ
0
j
;
j
=
2 e
m
Re ⁡
{
ψ
∗
(
− i
ℏ ∇
− 2
e
A
)
ψ
}
{\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {j} \;\;;\;\;\mathbf {j} ={\frac
{2e}{m}}\operatorname {Re} \left\{\psi ^{*}\left(-i\hbar \nabla -2e\mathbf {A} \right)\psi \right\}}
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:
α ψ
+ β
|
ψ
|
2
ψ =
0.
{\displaystyle \alpha \psi +\beta |\psi |^{2}\psi =0.\,}
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 > Tc. 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:
|
ψ
|
2
= −
α β
.
{\displaystyle |\psi |^{2}=-{\frac }.}
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 (T − Tc) with
α0/β > 0:
Above the superconducting transition temperature, T > Tc, 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 < Tc, the right hand side of the equation
above is positive and there is a non-trivial solution for ψ. Furthermore,
|
ψ
|
2
= −
α
0
( T
−
T
c
)
β
,
{\displaystyle |\psi |^{2}=-{\frac {\alpha _{0}(T-T_{c})}},}
that is ψ approaches zero as T gets closer to Tc 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. In this interpretation, |ψ|2 indicates the fraction
of electrons that have condensed into a superfluid.
== Coherence length and penetration depth ==
The Ginzburg–Landau equations predicted two new characteristic lengths in a superconductor
which was termed coherence length, ξ. For T > Tc (normal phase), it is given by
ξ =
ℏ
2
2 m
|
α
|
.
{\displaystyle \xi ={\sqrt {\frac {\hbar ^{2}}{2m|\alpha |}}}.}
while for T < Tc (superconducting phase), where it is more relevant, it is given by
ξ =
ℏ
2
4 m
|
α
|
.
{\displaystyle \xi ={\sqrt {\frac {\hbar ^{2}}{4m|\alpha |}}}.}
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
λ =
m
4
μ
0
e
2
ψ
0
2
,
{\displaystyle \lambda ={\sqrt {\frac {m}{4\mu _{0}e^{2}\psi _{0}^{2}}}},}
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 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 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 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.
== 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, 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. Gaiotto, Gukov & Seiberg (2013)
== See also