ScalarQuantumElectrodynamics - Freie Universitätusers.physik.fu-berlin.de/~kleinert/b6/psfiles/Chapter-16-scalarql.pdf · ScalarQuantumElectrodynamics ... In that form it is known

A complex system that works is invariably found

to have evolved from a simple system that works.

John Gall

17

Scalar Quantum Electrodynamics

In nature, there exist scalar particles which are charged and are therefore coupled tothe electromagnetic field. In three spatial dimensions, an important nonrelativisticexample is provided by superconductors. The phenomenon of zero resistance at lowtemperature can be explained by the formation of so-called Cooper pairs of electronsof opposite momentum and spin. These behave like bosons of spin zero and chargeq = 2e, which are held together in some metals by the electron-phonon interaction.Many important predictions of experimental data can be derived from the Ginzburg-

Landau theory of superconductivity [1]. The relativistic generalization of this theoryto four spacetime dimensions is of great importance in elementary particle physics.In that form it is known as scalar quantum electrodynamics (scalar QED).

17.1 Action and Generating Functional

The Ginzburg-Landau theory is a three-dimensional euclidean quantum field theorycontaining a complex scalar field

ϕ(x) = ϕ1(x) + iϕ2(x) (17.1)

coupled to a magnetic vector potential A. The scalar field describes bound statesof pairs of electrons, which arise in a superconductor at low temperatures due to anattraction coming from elastic forces. The detailed mechanism will not be of interesthere; we only note that the pairs are bound in an s-wave and a spin singlet stateof charge q = 2e. Ignoring for a moment the magnetic interactions, the ensemble ofthese bound states may be described, in the neighborhood of the superconductivetransition temperature Tc, by a complex scalar field theory of the ϕ4-type, by aeuclidean action

AE =∫

d3x

[

1

2∇ϕ∗

∇ϕ+m2

2ϕ∗ϕ+

g

4(ϕ∗ϕ)2

]

. (17.2)

The mass parameter depends on the temperature like m2 ∝ (T − Tc)/Tc. Abovethe transition temperature, the parameter m2 is positive, below it is negative. As

1077

1078 17 Scalar Quantum Electrodynamics

a consequence, the system exhibits a spontaneous symmetry breakdown discussedin Chapter 16. The symmetry group that is broken, is given by the U(1) phasetransformation

ϕ(x) → e−iαϕ(x) (17.3)

under which AE is obviously invariant. Alternatively we may write

ϕ1 → cosαϕ1 + sinαϕ2 ,

ϕ2 →− sinαϕ1 + cosαϕ2 , (17.4)

so that we may equally well speak of an O(2)-symmetry.The action (17.2) is the euclidean version of a relativistic field theory inD = 2+1

dimensions, with an action

AE =∫

dDx

[

1

2∂µϕ

∗∂µϕ− m2

2ϕ∗ϕ− g

4(ϕ∗ϕ)2

]

. (17.5)

For the sake of generality, we shall discuss this theory for arbitraryD. Most formulaswill be written down explicitly in D = 4 dimensions, to emphasize analogies withproper QED discussed in Chapter 12.

The phenomenon of spontaneous symmetry breakdown in this system has beenstudied in detail in Chapter 16. For T > Tc, where the mass term is positive, smalloscillations around ϕ = 0 consist of two degenerate modes carried by ϕ1, ϕ2, bothof mass m2. For T < Tc, where the mass is negative, the energy is minimized by afield ϕ with a real non-vanishing vacuum expectation value, say

〈ϕ(x)〉 ≡ ϕ0. (17.6)

Then the symmetry between ϕ1 and ϕ2 is broken and there are two different modesof small oscillations: one orthogonal to 〈ϕ〉, which is the massless Nambu-Goldstonemode, and one parallel to 〈ϕ〉, which has a positive mass −2m2 > 0.

In order to describe the phenomena of superconductivity with the euclideanversion of (17.5), we must include electromagnetism. According to the minimalsubstitution rules described in Chapter 12, we simply replace ∂µϕ in (17.5) by thecovariant derivative

∂µϕ→ Dµϕ ≡ (∂µ − iqAµ)ϕ. (17.7)

The extra vector potential Aµ(x) is assumed to be governed by Maxwell’s electro-magnetic action [see (12.3)]:1

Aγ = −1

4

∫

dDxFµν(x)Fµν(x). (17.8)

The combined action becomes

Ase[ϕ, ϕ∗, A] =

∫

dDx

{

1

2|Dµϕ|2 −

m2

2|ϕ|2 − g

4|ϕ|4 − 1

4F 2µν

}

. (17.9)

1In this chapter we use natural units with h = c = 1.

17.1 Action and Generating Functional 1079

where F 2µν is short for FµνF

µν . This action is invariant under local gauge transfor-mations

ϕ(x) → eiqΛ(x)ϕ(x), Aµ(x) → Aµ(x) + ∂µΛ(x). (17.10)

A functional integration leads to the generating functional of scalar QED:

Z [λ, λ∗, j] =∫

DϕDϕ∗∫

phys.DAµei{Ase[ϕ,ϕ∗,Aµ]−

∫

xλ∗ϕ−

∫

xϕ∗λ−

∫

xjµAµ}, (17.11)

where the symbol∫

x abbreviates the volume integral∫

d4x, and λ(x), λ∗(x) are thesources of the complex scalar fields ϕ∗(x) and ϕ(x). The vector jµ(x) is an externalelectromagnetic current coupled to the vector gauge field Aµ(x). To preserve thegauge invariance, the current is assumed to be a conserved quantity and satisfy∂µj

µ(x) = 0. The functional integration over the vector field∫

phys DAµ assumesthat a Faddeev-Popov gauge fixing factor is inserted, together with a compensatingdeterminant (recall the general discussion in Section 14.16):

∫

physDAµ ≡

∫

DAµΦ−1[A]F [A]. (17.12)

For m2 > 0, we may choose [recall (14.353)]

F [A] = F4[A] = e−12α

∫

x(∂µAµ)2 , (17.13)

and derive from (17.11) the Feynman rules for perturbation expansions. The gauge-fixed photon propagator was given in Eq. (14.373):

G0µν(x− y) = −∫

dDk

(2π)De−ik(x−y) i

k2

[

gµν − (1− α)kµkνk2

]

. (17.14)

We have seen in Chapter 12 that in QED all scattering amplitudes are independentof the gauge parameter α.

In addition, the perturbation expansion derived from the generating functional(17.11) employs propagators of scalar particles represented by lines

e−ik(x−y) G0(x− y) =∫

dDk

(2π)De−ik(x−y) i

k2 −m2 (17.15)

and vertices

. (17.16)

Since the scalar field is complex, the particle carries a charge, and the chargedparticle lines in the Feynman diagrams carry an orientation. The coupling of thescalar field to the vector potential does not only give rise to a three-point vertex ofthe type (12.95) in QED [left-hand diagram in (17.16)], but also to a four-vertex inwhich two photons are absorbed or emitted by a scalar particle [right-hand diagramin (17.16)]. This diagram is commonly referred to as seagull diagram.


17.2 Meissner-Ochsenfeld-Higgs Effect

The situation is drastically different for T < Tc where m2 < 0. At the classicallevel, the lowest energy state of the pure ϕ theory is reached at ϕ = ϕ0 with

|ϕ0| = φ0 =√

−m2/g, i.e.,

ϕ0 = φ0e−iγ0 =

√

−m2

ge−iγ0 . (17.17)

Now the photon part of the action (17.9) reads

Aγ =∫

dDx

(

−1

4F 2µν −

q2

2

m2

gAµ

2

)

. (17.18)

Extremization of this leads to the Euler-Lagrange equation

∂2Aµ − ∂µ∂A− q2m2

gAµ = 0. (17.19)

Note that we could have chosen as well

ϕ = φ0e−iγ0(x) =

√

−m2

qe−iγ0(x), (17.20)

with an arbitrary spacetime-dependent phase γ0(x). Then the photon Lagrangianwould read

Aγ =∫

dDx

[

−1

4F 2µν −

q2

2

m2

g(Aµ − ∂µγ0)

2

]

, (17.21)

with the Euler-Lagrange equation

∂2Aµ − ∂µ(∂A)− q2m2

g(Aµ − ∂µγ0) = 0. (17.22)

The extremal field is reached at

Aµ(x) = ∂µγ0(x). (17.23)

Of course, the two results (17.19) and (17.22) are equivalent by a gauge transforma-tion Aµ → Aµ + ∂µγ0(x), and the physical content of (17.21) is independent of theparticular choice of γ0(x).

Consider now the fluctuation properties of this theory. The generating func-tional has the form (17.11) in which the measure sums over all ϕ and physical Aµ

configurations orthogonal to the gauge degrees of freedom. If we decompose ϕ intoradial and azimuthal parts

ϕ(x) = ρ(x)e−iγ(x), (17.24)

17.2 Meissner-Ochsenfeld-Higgs Effect 1081

the path integral takes the form

Z [λ, λ∗, j] =∫

ρDρDγ∫

physDAµei{Ase[ϕ,ϕ∗,Aµ]−

∫

xλ∗ϕ−

∫

xϕ∗λ−

∫

xjµAµ}. (17.25)

But from the structure of the covariant derivative it is obvious that no matter whatthe fluctuating γ(x) configuration is, it can be absorbed into A(x) as a longitudi-nal degree of freedom. Thus we have the option of considering complex fields ϕ(x)together with physical transverse Aµ fluctuations, or real ρ together with trans-verse and longitudinal Aµ(x)-fluctuations. This may be expressed by rewriting thefunctional integral (17.25) as

Z [λ, λ∗, j] =∫

ρDρ∫

allDAµei{Ase[ρ,Aµ]−

∫

xλ∗ϕ−

∫

xϕ∗λ−

∫

xjµAµ}, (17.26)

with the action

A[ρ, Aµ] =∫

dDx

[

1

2(∂µρ)

2 +q2

2ρ2Aµ

2 − m2

2ρ2 − g

4ρ4 +

1

4Fµν

2

]

. (17.27)

Due to the second term in the exponent, the formerly irrelevant gauge degree offreedom in the vector potential Aµ(x) has now become a physical one.

Let us describe the situation more precisely in terms of a gauge-fixing func-tional F [A] of (17.13) and a compensating Faddeev-Popov determinant Φ−1[A] ofEq. (14.354). Whereas all former gauge conditions involve only the Aµ(x)-field itself,this is no longer true here. Now the transformations of Aµ and ϕ are coupled witheach other by a gauge transformation which involves the two fields (or any otherfield of the system). The gauge condition will be some functional

F [Aµ, ϕ] = 0, (17.28)

which has to be compensated by some functional Φ[Aµ, ϕ] depending on both fields.The latter is determined by the integral over the gauge volume as in (14.354):

Φ[Aµ, ϕ] =∫

DΛF[

AΛ, ϕΛ]

. (17.29)

It is most conveneiant to choose a gauge condition which ensures the reality of thecomplex scalar field ϕ via the δ-functional:

F [Aµ, ϕ] = δ[ϕ2(x)], (17.30)

where ϕ2(x) = Imϕ(x). Since the gauge transformations change

ϕ→ e−iΛ(x)ϕ = [cosΛ(x)ϕ1 + sinΛ(x)ϕ2] + i [− sin Λ(x)ϕ1 + cosΛ(x)ϕ2] , (17.31)

we see that

Φ[Aµ, ϕ] =∫

DΛ δ [− sin Λϕ1 + cos Λϕ2] . (17.32)


The result must be independent of the choice of gauge and can therefore be obtainedat ϕ = (|ϕ|, 0), where we find

Φ[Aµ, ϕ] =1

|ϕ(x)| . (17.33)

Note that Φ involves no derivative terms as in the earlier gauges. For this reasonit does not require Faddeev-Popov ghost fields. Setting ϕ = ρ(x)e−iγ(x), we arriveprecisely at the functional integral (17.27). The Faddeev-Popov determinant Φ =ρ(x) turns out to be responsible for bringing the integral at each point to a formexpected from the radial-azimuthal decomposition ϕ = ρe−iγ , and the associatedintegration measure dϕdϕ∗ = dϕ1dϕ4 = ρdρdγ.

The generating functional (17.26) may now be used for perturbation expansionsbelow Tc. Inserting into (17.27)

ρ(x) = ρ0 + ρ′(x) =

√

−m2

g+ ρ′(x), (17.34)

we can rewrite

A =∫

dDx

[

1

2(∂µρ

′)2 − (−2m2)

2ρ′2 − g

4ρ′4 − 3

2gρ0ρ

′3

−1

4Fµν

2 +µ2

2A2

µ + q2ρ0ρ′Aµ +

q2

2ρ′2Aµ

2

]

, (17.35)

with

µ2 ≡ −q2m2

g. (17.36)

This is the mass by which the Nambu-Goldstone theorem is violated. The model ex-hibits a spontaneous breakdown of the continuos U(1)-symmetry which by Noether’stheorem should produce a massless excitation. But the mixing of this excitation withthe field of the massless electromagnetic field has absorbed the would-be Nambu-Goldstone boson into the longitudinal part of the vector field, making it massive.This process is described more vividly by saying that the photon has “eaten up” thewould-be Nambu-Goldstone boson and grown “fat”.

The same mechanism has been used to unify the theories of electromagnetic andweak interactions. Then the scalar field ϕ possesses more components than the realand imaginary parts of the complex field ϕ, and the associated Lagrangian has ahigher symmetry than U(1). In that case the analog to the ρ-mass

m2ρ =

−2m2

2(17.37)

is referred to as the Higgs mass [see Eq. (27.99)], and the mass (17.36) of the fatvector field is referred to as the vector boson mass. This will be discussed in detailin Chapter 27.


From the quadratic part in ρ′ we deduce the massive ρ′ propagator

ρ′(x)ρ′(0) =∫

dq

(2π)4e−iqx i

q2 − (−2m2) + iη. (17.38)

The ρ′-field has cubic ρ′3- and quadratic ρ′4-interactions. The originally masslessGoldstone field γ has disappeared. It has been “eaten up” by the photon, makingit massive, and giving it one more polarization degree of freedom.

In the action (17.35) there are electromagnetic vertices in which one or twoparticles couple to two photons

q2ρ0 . (17.39)

The most important new feature is found in the propagator of the photon field.Given the quadratic photon part of the action in momentum space

A = −1

2

∫

dDk

(2π)4Aµ(−k)

[

kµkν − gµν(

k2 − µ2)]

Aν(k), (17.40)

where the kinetic matrix between the fields is

Mµν = kµkν − gµν(

k2 − µ2)

(17.41)

= −(

gµν − kµkν

k2

)

(

k2 − µ2)

+ µ2kµkν

k2. (17.42)

By inverting this we find the matrix

M−1µν = Gµν(k) = −g

µν − kµkν/k2

k2 − µ2+

1

µ2

kµkν

k2= −g

µν − kµkν/µ2

k2 − µ2(17.43)

so that the massive propagator becomes

Gµν(x) =∫

dk

(2π)4e−ikxGµν(k) =

∫

dk

(2π)4e−ikx −i

k2 − µ2

(

gµν −kµkνk2

)

. (17.44)

The photon has acquired a mass and can no longer propagate over a long range.The dramatic experimental consequence is known as the Meissner-Ochsenfeld effect[2]: If a superconductor is cooled below Tc, the magnetic field lines are expelled andcan invade only a thin surface layer of penetration depth 1/µ.

At k2 = µ2, there is a massive particle pole. Since the propagator Gµν(k) at thispole is purely transverse, it satisfies

kµGµν(k) = 0. (17.45)

The massive particles have three internal degrees of freedom corresponding to thethree orientations of their spin. This has to be contrasted with the massless gaugeinvariant theory for T > Tc, where gauge invariance permits to choose, say, A0 = 0


while still satisfying (17.45) at the pole, thereby eliminating one more degree offreedom in the asymptotic states. Only two transverse photon polarizations remainfar away from the interaction. There is no contradiction in the number of degrees offreedom since for T > Tc there are two modes carried by the complex ϕ fields. Thethird photon degree is a consequence of absorbing, into the vector field, the phaseoscillations, i.e., the former Nambu-Goldstone degrees of freedom.

Certainly, the same theory can also be calculated with the earlier gauge fixingLagrangian

LGF = − 1

2α(∂µAµ)

2 , (17.46)

in which case the quadratic piece of the action is

Aquad =∫

dDx

[

1

2(∂µρ

′)2 − (−2m2)

2ρ′2 (17.47)

−1

4Fµν

2 +µ2

2

(

Aµ −1

q∂µγ

)2

− 1

2α(∂µA

µ)2

.

Recalling the Faddeev-Popov determinant Φ−14 of (14.365), we have to add for this

gauge the Faddeev-Popov ghost action corresponding to (14.418):

Aghost = − 1

2α

∫

dDx ∂µc∗∂µc . (17.48)

The generating functional is obtained by integrating the exponentialei{Ase+Aghost−λ∗ϕ−ϕ∗λ−jµAµ} over all Aµ(x), all complex ϕ(x), and all ghost config-urations c∗(x), c(x):

Z [λ, λ∗, j] =∫

DcDc∗∫

DϕDϕ∗∫

DAµ ei{Ase+Aghost−λ∗ϕ−ϕ∗λ−jµAµ}. (17.49)

What is the particle content of this theory? Since the Faddeev-Popov ghosts donot interact with the other fields in the action, they can be ignored. As far as thephoton is concerned, let us split the vector potential as

Aµ = Atµ + Al

µ, (17.50)

where At is the transverse part satisfying the Lorenz gauge condition ∂µAtµ = 0, and

Al is the longitudinal part Alµ = ∂−2∂µ∂νA

ν , which is a pure gauge field.2 Then thetransverse part has the propagator

Atµ(x)A

tν(0) =

∫

dDk

(2π)D−i

k2 − µ2

(

gµν −kµkνk2

)

. (17.51)

2As in (14.370), the letters l and t stand for longitudinal and transverse to the four -vector fieldAµ(x).


The gauge-like pieces, on the other hand, are coupled via the would-be Nambu-Goldstone mode. In terms of the pure gradient field Aµ = kµΛ, and using the scalarfield γ′ ≡ (1/q) γ, we find the quadratic Lagrange density in energy-momentumspace

µ2

2(Λ, γ′)

(

k2 − α−1k4 −k2− k2 k2

)(

Λγ′

)

, (17.52)

which has the determinant −µ4/4αk6, and eigenvalues

λk =µ2k2

2

1− 1

2α

k2

µ2±

√

√

√

√1 +

(

k2

2αµ2

)2

. (17.53)

Both vanish at k2 = 0 (and only there). Thus there are two more massless asymp-totic states in Hilbert space. At first sight, this formulation describes a physicalsituation different from the previous one. We do know, however, that the gener-ating functionals are the same. In the previous formulation, there were only twotypes of particles. Therefore all additional zero-mass asymptotic states found inthe present gauge must be such that, if a scattering process takes place with thetwo physical particles coming in, the unphysical ones can never be produced. Phys-ical and unphysical states should have no mutual interaction. We may say thatthe S-matrix is irreducible on the space of physical states. For the Faddeev-Popovghost fields this is trivial to see. With respect to the other states, however, it is notso obvious, and the path integral is the only efficient tool for convincing us of thecorrectness of this statement, as demonstrated in Section 14.16. The irreducibilityof the S-matrix becomes most transparent by using a particular gauge due to ’tHooft in which physical and unphysical parts of the Hilbert space receive a cleardistinction which depends on the gauge parameter. Consider the gauge condition

F [A,ϕ] = exp

− 1

2α

(

∂µAµ − µ2

qαγ

)2

. (17.54)

For α → 0 this enforces the Lorenz gauge. The exponent has the pleasant advantagethat, when added to the free part of the Lagrangian a term mixing γ with the gauge-like components of Aµ disappears, so that no further diagonalization is requiredin (17.52). Therefore, postponing the calculation of Φ−1[A,ϕ] to Eq. (17.60), thegauge-fixed free action reads

Afree =∫

dDx

{

1

2(∂ρ′)

2 − −2m2

2ρ′2 − 1

4Fµν

2 +µ2

2A2

µ

+µ2

2q2

[

(∂µγ)2 − αµ2γ2

]

− 1

2α(∂µA

µ)2}

. (17.55)

Thus the photon has a quadratic piece

1

2Aµ(−k)

[

−(

k2 − µ2)

gµν + kµkν −1

αkµkν

]

Aν(k)


=1

2Aµ(−k)

[

−(

k2 − µ2)

(

qµν −kµkνk2

)

− k2 − αµ2

α

kµkν

k2

]

Aν(k), (17.56)

which by inversion of the matrix between the fields leads to a propagator

Gµν(x) =∫ dDk

(2π)4e−ikx

[

− i

k2 − µ2

(

gµν −kµkνk2

)

− iα

k2 − αµ2

kµkν

k2

]

. (17.57)

Apart from the physical propagator, there is a ghost-like state in Aµ(k) parallel tokµ.

The new feature of this gauge is that the would-be Nambu-Goldstone mode γ(x)has acquired a mass term αµ2γ2(x). The reason for this lies in the fact that thegauge fixing term destroys explicitly also the global invariance of the Lagrangian

ϕ(x) → e−iΛϕ(x), γ(x) → γ(x) + Λ, Aµ → Aµ, (17.58)

under which the gauge pieces transform as

− 1

2α

(

∂µAµ − µ2

qαγ

)2

→ − 1

2α

[

∂µAµ − µ2

qα(γ + Λ)2

]

. (17.59)

Therefore, pure phase transformations change the energy, and this results in a massterm.

Let us now take a look at the functional Φ[A,ϕ] in order to see that this doesnot modify the Lagrangian (17.55). According to (14.354), we have to integrate

Φ[A,ϕ] =∫

DΛ exp

− 1

2α

(

∂µAΛµ − µ2

qαγΛ

)2

=∫

DΛ exp

− 1

2α

(

∂µAµ + ∂2Λ− µ2α

q(γ + Λ)

)2

(17.60)

=∫

DΛ exp

− 1

2α

(

∂µAµ −µ2

qαγ +

(

∂2 − µ2

qα

)

Λ

)2

which, after a shift of the integration variable, gives the trivial constant

Φ[A,ϕ] = det

(

−∂2 + µ2

qα

)

/√α. (17.61)

This can be expressed in terms of Faddeev-Popov ghost fields as

Φ−1[A,ϕ] =∫

DcDc∗ exp[

i√α

∫

dx

(

∂c∗∂c +µ2

qα c∗c

)]

. (17.62)

The situation is similar to that of the last gauge condition: There are three additionalkinds of asymptotic states: Faddeev-Popov ghosts, ghost-like poles in the gauge part

17.3 Spatially Varying Ground States 1087

of the photon field and a state carried by the would-be Nambu-Goldstone field. Butcontrary to the previous case, all these states now have a mass which depends onthe gauge parameter α. This proves that they must be artifacts of the gauge choiceand cannot contribute to physical processes. In particular, by letting the parameterα tend to infinity, all these particles become infinitely heavy and are thus eliminatedfrom the physical Hilbert space.

17.3 Spatially Varying Ground States

The theory of a complex scalar field interacting with a photon has another interestingfeature. The ground state must not necessarily be uniform. There are certain rangesof mass and external source where the system prefers to be filled with string-likefield configurations, and others where the fields have spatially periodic structures ofthe hexagonal type. In order to see how this comes about and to be able to compareresults with experimental data we consider the euclidean three-dimensional scalarQED, which is the Ginzburg-Landau theory of superconductivity [1]. Superconduc-tivity arises if electrons in a crystal are attracted so strongly by the effects of theelastic forces that the attraction overcomes the Coulomb repulsion. Then they formCooper pairs which can form a condensate very similar to the bosonic helium atoms.In the absence of electric fields, the mean-field approximation to the effective actionis

1

TΓ[Φ] = −

∫

d3x

[

1

2H2 +

1

2(DΦ)∗DΦ+

m2

2|Φ|2 + g

4|Φ|4

]

. (17.63)

Since the charge carriers are electron pairs, the charge of the field is twice theelectronic charge, and the covariant derivative (17.7) becomes

D = ∇− iqA = ∇− i2e

hcA. (17.64)

The first expression is written down in natural units with h = c = 1, the second inphysical CGS-units. The curl of the vector potential is the magnetic field:

H(x) = ∇×A(x). (17.65)

The time-independent equations of motion are

(−i∇ − qA)2Φ+m2Φ + g|Φ|2Φ = 0, (17.66)

and the curl of the magnetic field yields

∇×H = j, (17.67)

where j is the current density of the matter field

j =q

2iΦ∗ ↔

∇Φ− q2|Φ|2A. (17.68)


In order to describe superconductivity in the lower neighborhood of the criticaltemperature, we insert a temperature dependent mass term

m2 = µ2(

T

Tc− 1

)

= −µ2τ, (17.69)

thereby choosing the minus sign to focus upon the regime below Tc.

17.4 Two Natural Length Scales

Before proceeding it is useful to introduce reduced field quantities and define

ψ =Φ

|Φ0|, (17.70)

with

|Φ0| =√

−µ2

g=

µ√g

√τ (17.71)

being the nonzero vacuum expectation of the ϕ-field for T < Tc. We also define alength scale associated with the massive fluctuations of the Φ-particle

ξ(τ) =1

µ√2. (17.72)

A second length scale charecterizes the range of the photon, after it has “eaten up”the Nambu-Goldstone boson:

λ(τ) =1

q|Φ0|=

√g

qµ√τ=

√g

qξ(τ). (17.73)

The first length scale is usually referred to as coherence length, the second as pen-etration depth, for reasons which will become plausible soon. The ratio of the twolength scales is an important temperature-independent material constant, called theGinzburg-Landau parameter and denoted by κ:

κ ≡ λ(τ)

ξ(τ)=

√g

q, (17.74)

which measures the ratio between the coupling strength g with respect to q2. Thecoherence length may be used to introduce a reduced dimensionless vector potential

a ≡ qξ(τ)A (17.75)

and a reduced magnetic field

h = κ∇× a = qκλ(τ)∇×A. (17.76)

17.4 Two Natural Length Scales 1089

In the absence of a magnetic field, the action may then be expressed as

1

TΓ [Φ] = −4fc ξ

3F , (17.77)

where fc is the so-called condensation energy density:

fc =Φ2

0m2

4= −m

4

4g= −µ

4

4gτ, (17.78)

and F is the reduced free energy

F =1

ξ3

∫

d3x[

1

2|ξ(τ)(∇− ia)ψ|2 − 1

2|ψ|2 + 1

4|ψ|4 + 1

2λ2(τ)h2

]

. (17.79)

Note that h measures the magnetic field in units of

H0 =√2Hc ≡

1

qλξ=√

m2Φ20 = 2

√

fc, (17.80)

where Hc is the value at which the magnetic energy density H2c /2 equals the con-

densation energy density fc.If we agree to measure all lengths x in multiples of the coherence length ξ, the

reduced free energy F becomes simply

F =∫

d3x[

1

2|(∇− ia)ψ|2 − 1

2|ψ|2 + 1

4|ψ|4 + h2

]

. (17.81)

In the reduced variables, the equations of motion are simply

(i∇+ a)2 ψ = ψ − |ψ|2ψ, (17.82)

κ∇× h = κ2∇× (∇× a) = κ2[∇2a−∇ (∇ · a)] = j, (17.83)

where j is the current (17.68) in natural units

j =1

2iψ∗ ↔

∇ψ − |ψ|2a. (17.84)

Let us also write down these equations in the polar field decomposition

ψ(x) = ρ(x)eiγ(x). (17.85)

It is convenient to absorb the gradient of γ(x) into the vector potential, and to goover to the field

a → a−∇γ. (17.86)

This brings F to the simple form

F =∫

dx{

1

2|(∇− ia) ρ|2 − 1

2ρ2 +

1

4ρ4 + (∇× a)2

}

, (17.87)

and the field equations (17.82) and (17.83) become(

−∇2 + a2 − 1 + ρ2

)2ρ = 0, (17.88)

κ∇× h = κ2∇× (∇× a) = κ2[∇2a−∇ (∇ · a)] = −ρ2a. (17.89)

These will be most suitable for upcoming discussions.


17.5 Planar Domain Wall

As mentioned earlier, it has been known for a long time that superconductors havethe tendency of expelling magnetic fields (Meissner-Ochsenfeld effect) [2]. Let usstudy what the free energy (17.87) has to say about this magnetic phenomenon. Inorder to get some rough ideas it is useful to consider a material sample in a magneticfield that points in the x-direction. Let us allow for variations of the system onlyalong the z-coordinate. We may choose the vector potantial a0 to point purely intothe x-direction. If we denote the x-component of a(z) ≡ a0(z), the fields are

hy(z) = κa′(z), hx = hy = 0. (17.90)

The reduced free energy density reads

f =1

2ρ′2 − 1

2ρ2 +

1

4ρ4 + a2ρ2 + κ2a′2. (17.91)

Its extremization yields the equation of motion for ρ(z):

−ρ′′ + a2ρ = ρ− ρ3, (17.92)

and for a(z):

κ2a′′ = aρ2. (17.93)

Differentiating (17.93) and inserting (17.90), we obtain

κ2ρ2d

dz

(

1

ρ2dh

dz

)

= κ2(

h′′ − 2h′ρ′

ρ

)

= hρ2. (17.94)

Inserting (17.93) into (17.92) yields

−ρ′′ + κ2h′2

ρ3= ρ− ρ3. (17.95)

The last two equations can be used as coupled differential equations for h and ρ.If these two fields are known, the vector potential may be calculated from (17.93)

as

a =a′′

ρ2=h′

ρ2. (17.96)

This equation gives an immediate result: A constant magnetic field can exist onlyfor ρ = 0, i.e., if the system has no spontaneous symmetry breakdown. If we assume,conversely, that ρ = const. 6= 0, then Eq. (17.92) shows that also a2 is a constant,namely a2 = (1− ρ2) /ρ. From Eq. (17.96) we see that then also h′ is a constant.There is no contradiction if and only if ρ = 1, in which case (17.92) gives a = 0, and

17.5 Planar Domain Wall 1091

(17.96) enforces h′ = 0. After this, Eq. (17.94) shows that only h = 0 is a consistentsolution. Therefore, constant solutions have only the two alternatives

h = const 6= 0, ρ = 0, normal phase (17.97)

h = 0, ρ = 1, superconductive phase. (17.98)

The second alternative exhibits the experimentally observed Meissner-Ochsenfeldeffect [2], implying that the superconductive state does not permit the presence ofa magnetic field.

Consider now z-dependent field configurations. In order to simplify the discus-sion let us first look at two limiting situations:

Case I : κ≪ 1/√2

This corresponds to a very short penetration depth of the magnetic field which maypropagate only over length scales λ(T ) = κξ(τ) or, in natural units, z ∼ 1, i.e., ithas a unit range. For very small κ, Eq. (17.95) becomes

ρ′′ ≈ ρ(

1− ρ2)

, (17.99)

which can be integrated by multiplying it with ρ′ to find

−ρ′2 ≈ 1

2

(

1− ρ2)2

+ E. (17.100)

If the system is in the superconductive state for large z, the field satisfies the bound-ary condition corresponding to the alternative (17.97), such that ρ→ 1 for z → ∞.Inserting this into (17.100) we see that the constant of integration E must be zero.Integrating the resulting equation further gives

z =√2∫

dρ

1− ρ2=

√2Atanh ρ, (17.101)

or

ρ(z) = tanhz√2. (17.102)

The field configuration is displayed on the right-hand side of the upper part ofFig. 17.1.

For z < 0, the solution (17.102) becomes meaningless, since ρ > 0 by definition.We can continue the solution into this regime by matching it with the trivial solution(17.97):

h = const, ρ = 0, (17.103)

which is shown as the left-hand branch in the upper part of Fig. 17.1.The size of the constant magnetic field in the first case is determined by the

fact that the free energy density must be the same at large positive and negative z.


Otherwise, there would be energy flow. Inserting (17.100) for E = 0 into (17.91), itbecomes approximately

f ≈ −1

4+

1

2a2ρ2 +

1

2h2. (17.104)

For large z where h = a = 0 and ρ = 1, this is equal to −1/4. For large negative z,f is equal to h2. Hence the constant h in the normal phase is equal to

h = hc =1√2. (17.105)

Within the region around z ≈ 0, the magnetic field h drops to zero over a unit lengthscale determined by (17.93), which is very short compared with the length 1/κ overwhich ρ varies (this is precisely the coherence length). If we plot the transitionregion against z, the field h jumps abruptly to zero from h = hc while ρ has asmooth increase. The full approximate solution plotted in Fig. 17.1 is

h(z) ≈ hcΘ(−z), (17.106)

ρ(z) ≈ tanhz√2. (17.107)

H

N S

type I

type II

ρ

κ ≫ 1,

κ ≪ 1,

Figure 17.1 Dependence of order parameter ρ and magnetic field H on the reduced

distance z between normal and superconductive phases. The magnetic field points parallel

to the domain wall.

Case II : κ≫ 1/√2

Consider now the opposite limit of a very large penetration depth. Here we approx-imate (17.95) by

κ2h′2 ≈ ρ4(

1− ρ2)

, (17.108)

implying that we may omit the gradient term ρ′′ in Eq. (17.92):

a2ρ ≈ ρ(

1− ρ2)

. (17.109)

17.5 Planar Domain Wall 1093

Let us calculate ρ again by starting out with ρ = 1 at z = ∞. Then h must decreasefor positive z and we have to take the positive square-root

κh′ = ρ2√

1− ρ2. (17.110)

From Eq. (17.94), on the other hand, we obtain

h ≈ κ2d

dz

(

1

ρ2d

dzh

)

= −κ ddz

√

1− ρ2. (17.111)

It is convenient to introduce the abbreviation

u ≡√

1− ρ2, (17.112)

such that (17.111) becomes

h = −κu′, (17.113)

and (17.110) turns into the simple differential equation

κ2u′′ =(

1− u2)

u. (17.114)

After multiplying this with u′, and integrating, we find

κ2u′2 = u2(

1− u2

2

)

+ const. (17.115)

Imposing the asymptotic condition that, at large z, the order field ρ goes against 1,the function u(z) must vanish in this limit, fixing the constant to zero. The equationis then solved by

z − z0 = κ∫ du

u

1√

1− u2/2= κatanh

1√

1− u2/2(17.116)

or

u(z) =

√2

cosh[(z − z0)/κ]. (17.117)

From (17.112) we find

ρ =

√

1− 2

cosh2[(z − z0)/κ]. (17.118)

As z comes in from large positive values, ρ decreases. We can arrange it to becomezero at z = 0 if we choose z0 such that

sinh(z0/κ) = 1. (17.119)


From Eq. (17.113), the magnetic field is then:

h(z) =

√2 sinh[(z − z0)/κ]

cosh2[(z − z0)/κ]. (17.120)

It is zero at z = ∞, and becomes gradually stronger as it approaches z = 0, whereit reaches the value

h = hc =1√2, (17.121)

as before. From there on we may again match the trivial solution continuously bysetting

h ≡ hc, ρ ≡ 0; z < 0. (17.122)

The full solution shown in Fig. 17.1 is then

ρ(z) = Θ(z)

√

1− 2

cosh2[(z − z0)/κ], (17.123)

h(z) = Θ(−z)hc +Θ(z)

√2 sinh[(z − z0)/κ]

cosh2[(z − z0)/κ]. (17.124)

The ranges, over which h and ρ vary, are of equal order κ, or in physical units oforder λ = κξ.

Case III : κ = 1/√2

For this κ-value, a trial ansatz

h =1√2

(

1− ρ2)

(17.125)

can be inserted into (17.94), and leads to the second-order differential equation(

1− ρ2)

ρ2 = −ρρ′′ + ρ′2. (17.126)

This, in turn, happens to coincide with the other equation (17.95), thus confirmingthe correctness of the ansatz (17.125). For z = ∞, the fields start out with ρ = 1and h = 0, and go to ρ = 0, and h = hc = 1/

√2 for z → −∞.

Equation (17.126) takes a particularly appealing form if we consider the auxiliaryfunction σ(z) defined by ρ(z) ≡ eσ(z)/2. Then

ρ′ =1

2σ′eσ/2, ρ′′ =

1

2σ′′eσ/2 +

1

kσ′2eσ/2, (17.127)

and we may rewrite (17.126) as

− (1− eσ) =ρ′′

ρ− ρ′2

ρ2=

1

2σ′′. (17.128)

17.6 Surface Energy 1095

This can be integrated to

eσ − 1− σ =σ′2

4, (17.129)

which has the solution

z = −1

2

∫ σ

a

dσ′√eσ′ − 1− σ′ . (17.130)

For z → ∞, the solution σ(z) goes to zero like e−z/√2, such that ρ → 1. For

z → −∞, it becomes more and more negative like σ(z) ∼ −z2/4, such that ρ(z)goes to zero exponentially fast in z.

For intermediate values κ, the solutions have to be found numerically. They alllook qualitatively similar. The ratio κ of penetration depth to coherence length de-termines how far the magnetic field invades into the superconductive region relativeto the coherence length.

17.6 Surface Energy

Consider the energy per unit length for these classical fields as they follow fromEq. (17.91). Inserting the equations of motion and performing one partial integrationrenders the much simpler expression for the energy per area A, both in natural units:

F

A=∫

dzf =∫

dz(

−1

4ρ4 +

1

2h2)

. (17.131)

Obviously, the classical solution with an absolute minimum is h = 0, ρ = 1. Inorder to enforce the previously discussed configurations, an external magnetic fieldis needed, and we must minimize the reduced magnetic enthalpy per unit length area

G

A=∫

dz[

−1

4ρ4 +

1

2h2 − h hext

]

, (17.132)

where we have subtracted a term

mhext = h hext, (17.133)

in which the reduced magnetization m ≡ ∂hG/A = h is coupled to the reducedexternal magnetic field. Such a term does change the differential equations (17.92)and (17.93) for a and ρ, since it is a pure surface term hext∂za.

For z → −∞, where ρ→ 0 and h→ hc, the enthalpy goes asymptotically against

G

A≈∫

dz[

1

2h2c − hc hext

]

. (17.134)

For z → ∞, on the other hand, where the size of the order field goes to 1 and h→ 0,the asymptotic value is

G

A≈∫

dz(

−1

4

)

. (17.135)


Thus both asymptotic regimes have the same magnetic enthalpy for the particularexternal field hext = hc = 1/

√2. If the energy would be smaller in one regime, the

wall between superconductive and normal phase would start moving towards thatside, such as to decrease the energy. This would go on until the system is uniform.Thus we conclude: For h > hc, the system is uniformly normal, for h < hc it isuniformly super-conductive.

We now calculate the energy stored in the finite region around z = 0 at thecritical magnetic field hext = hc. There the density deviates only slightly from theasymptotic form, and we must evaluate the expression

G− Gas

A=∫ ∞

−∞dz

(

−1

4ρ4 +

1

2h2 − 1√

2h +

1

4

)

. (17.136)

Note the special properties of the case κ = 1/√2: Inserting (17.125), the surface

energy is seen to vanish.When inserting the κ≫ 1 -solutions (17.123) and (17.124) into (17.136), we find,

using Eqs. (17.112) and (17.113), the negative energy

G− Gas

A=

∫ ∞

−∞dz

u2(

1− u2

2

)

− 1√2u

(

1− u2

2

)1/2

=∫ ∞

−∞dz

u

√

1− u2

2− 1√

2

du

dz=(

−2

3

√2− 1

2

)

< 0. (17.137)

And the same sign holds for all κ > 1/√2.

For κ ≪ 1, on the other hand, the enthalpy (17.136) vanishes in the supercon-ducting phase for z > 0, where ρ = 1 and h = 0. In the normal phase, where h = hcby Eq. (17.105), we find

G− Gas

A=

∫ 0

−∞dz

1

4

(

1− ρ4)

=1

2√2

∫ 1

0dρ (1 + ρ2), (17.138)

which is positive for all κ < 1/√2.

Thus we conclude: In superconductors with κ > 1/√2, it is energetically more

favorable to form a wall in which the magnetic field develops from zero up to hc,instead of a uniform field configuration. For κ < 1/

√2, the opposite is true. The

first case is referred to as a soft or type-II superconductor, the second as a hardtype-I superconductor.

17.7 Single Vortex Line and Critical Field Hc1

Consider now a type-II superconductor in a small external magnetic field Hext, whereit is in the state of a broken symmetry with a uniform order field Φ = Φ0. Whenincreasing Hext, there will be a critical value Hc1 where the field lines first begin toinvade the superconductor. We expect this to happen in the form of a quantized

17.7 Single Vortex Line and Critical Field Hc1 1097

magnetic flux tubes. When increasing Hext further, more and more flux tubes willperforate the superconductor, until the critical magnetic field Hc is reached, wherethe sample becomes normally conducting. The regime between Hc1 and Hc is calledthe mixed state of the type-II superconductor.

The quantum of flux carried by each flux tube is

Φ0 =ch

2e≈ 2× 10−7 gauss cm2 . (17.139)

Such a flux tube may be considered as a line-like defect in a uniform superconductor.It forms a vortex line of supercurrent, very similar to the vortex lines in superfluidhelium discussed in Section 16.4. The two objects possess, however, quite differentphysical properties, as we shall see.

Suppose the system is in the superconductive state without an external voltageso that there is no net-current j. Suppose a flux tube runs along the z-axis. Thenwe can use the current (17.84) to find the vector potential

a = − j

|ψ|2 +1

2i

1

|ψ|2ψ† ↔∇ψ. (17.140)

Far away from the flux tube, the state is undisturbed, the current j vanishes, andwe have the relation

a =1

2i

1

|ψ|2ψ† ↔∇ψ. (17.141)

In the polar decomposition of the field ψ(x) = ρ(x)eiγ(x), the derivative of ρ(x)cancels, and a(x) becomes the gradient of the phase of the order parameter:

a(x) = ∇iγ(x). (17.142)

Here we can compare the discussion with that of vortex lines in superfluid helium inSection 16.4. There the superflow velocity was proportional to the gradient of thephase angle variable γ(x). The periodicity of γ(x) led to the quantization rule that,when taking the integral over dγ(x) along a closed circuit around the vortex line,it had to be an integral multiple of 2π [recall Eq. (16.107)]. The same rule applieshere:

∮

Bdγ(x) =

∮

Bdx ·∇γ(x) = 2πn. (17.143)

By Stokes’ theorem, this is equal to the integral∫

dS ·∇× a, where dS is a surfaceelement of the area enclosed by the circuit. Since h = κ(∇ × a) is the magneticfield in natural units [recall (17.86)], the integral (17.143) is directly proportional tothe magnetic flux through the area of the circuit

� =∫

SB

dS · h = κ∫

dS · (∇× a) = κ∮

Bdx · a = κ

∮

Bdx ·∇γ = 2πnκ. (17.144)


This holds in natural units, indicated by a bar on top of �, in which h = c = 1.The vector potential is given by (17.75), and x is measured in units of the coherencelength.

The quantization condition in physical CGS-units follows from the same argu-ment, after it is applied to the original current (17.68). Remembering the equality

q =2e

hc(17.145)

if Eq. (17.64) is expressed in CGS-units, the relation (17.141) reads

A =2e

2ihc

1

|ψ|2ψ† ↔∇ψ, (17.146)

and (17.142) becomes

A(x) =hc

2e∇iγ(x) =

�0

2π∇iγ(x). (17.147)

The magnetic flux integral in CGS-units is therefore

� =∫

SB

dS ·H =∫

dS · (∇×A) =∮

Bdx ·A =

Φ0

2π

∮

Bdx ·∇γ = n�0, (17.148)

and thus an integer multiple of the fundamental flux (17.139).It is instructive to perform the circuit integral (17.144) once more around a circle

close to the flux tube, where the current in Eq. (17.140) no longer vanishes. Theangular integral

∮

dx ·∇γ still has to be equal to 2πn, and we find the quantizationrule

∮

Bdx ·

(

A+j

|ψ|2)

= 2πn, (17.149)

or

�+1

|ψ|2∮

Bdx · j = 2πnκ. (17.150)

This shows that a smaller circuit, which encloses fewer magnetic field lines, containsan increasing contribution of the supercurrent around the center of the vortex line.The sum of the two contributions in (17.150) remains equal to 2πnκ. This showsthat the flux tube is also a vortex line with respect to the supercurrent. The circularcurrent density must be inversely proportional to the distance. This behavior willbe derived explicitly in Eq. (17.192).

Quantitatively, we can find the properties of a vortex line by solving the fieldequations (17.88) and (17.89) in cylindrical coordinates. Inserting the second intothe first equation, we find

−1

r

d

drrdρ

dr+κ2

ρ3

(

dh

dr

)2

− (1− ρ2)ρ = 0, (17.151)


where h denotes the z-component of h. Forming the curl of the second equationgives the cylindrical analogue of (17.96), i.e.,

h = κ21

r

d

dr

r

ρ2d

drh. (17.152)

For r → ∞, we have the boundary condition ρ = 1, h = 0 (superconductive statewith Meissner effect [2]) and j = 0 (no current). Since j is proportional to ∇×h byEq. (17.83), the last condition implies that

h′(r) = 0, r → ∞. (17.153)

In cylindrical coordinates, the flux quantization condition (17.144) can be writtenin the form

� = 2π∫ ∝

0dr r h(r) = 2πnκ. (17.154)

Inserting here Eq. (17.152) yields

� = 2πκ2[

r

ρ2h′(r)

]∞

0

= −2πκ2[

r

π2h′(r)

]

r=0, (17.155)

so that the quantization condition turns into a boundary condition at the origin:

h′ → −ρ2nκ

1

r, for r → 0. (17.156)

Inserting this condition into (17.151) we see that, close to the origin, ρ(r) satisfiesthe equation

−1

r

d

drrd

drρ(r) +

n2

r2ρ− (1− ρ2)ρ ≈ 0, (17.157)

implying the small-r behavior of ρ(r):

ρ(r) = cn

(

r

κ

)n

+O(

rn+1)

. (17.158)

Reinserting this back into (17.156) we have

h(r) = h(0)− c2n2κ

(

r

κ

)2n

. (17.159)

For large r, where ρ→ 1, the differential equation (17.152) is solved by the modifiedBessel function K0, with some proportionality factor α:

h(r) → αK0

(

r

κ

)

, r → ∞. (17.160)


More explicitly, the limit is

h(r) → α√

πκ/2re−r/κ., r → ∞. (17.161)

Consider now the deep type-II regime where κ≫ 1/√2. There ρ(r) goes rapidly

to unity, as compared to the length scale over which h(r) changes, which is κ in ournatural units. Therefore, the behavior (17.160) holds very close to the origin. Wecan determine α by matching (17.160) to (17.156) from which we find (using thesmall-r behavior K ′

0(r) = −K1(r) ≈ −l/r):

α =n

κ. (17.162)

In general, h(r) and ρ(r) have to be evaluated numerically. A typical solution forn = 1 is shown in Fig. 17.2 for κ = 10.

Figure 17.2 Order parameter ρ and magnetic field h for an n = 1 vortex line in a deep

type-II superconductor with K = 10.

The energy of a vortex line is obtained from Eq. (17.87). Inserting the equationsof motion (17.88) and (17.89), and subtracting the condensation energy, we obtainfrom (17.131)

∆Fvort = Fvort − Fc =1

2

∫

d3x[

1

2(1− ρ4) + h2

]

. (17.163)

For κ ≫ 1/√2, we may neglect the small radius r ≤ 1, over which ρ increases

quickly from zero to its asymptotic value ρ = 1. Above r ≈ 1 the magnetic field forr ≤ κ is given by (17.160). Inserting this into (17.151) with (17.158), we find

ρ(r) ≈ 1− n2

2r2. (17.164)

Thus the region 1 ≤ r ≤ κ yields an energy per unit length

1

LFvort =

1

22π∫ κ

1drr

[

1

2(1− ρ4) + h2

]

= πn2∫ κ

1drr

[

1

r2+

1

κ2K2

0

(

r

κ

)]

. (17.165)


For κ→ ∞, the second integral becomes a constant, as a consequence of the integral∫∞0 dx xK2

0 (x) = 1/2. The first integral, on the other hand, has a logarithmicdivergence, so that we find

1

LFvort ≈ πn2[log κ+ const.]. (17.166)

A more careful evaluation of the integral yields πn2(log κ+ 0.08).Let us now see at which external magnetic field such vortex lines can form. For

this we consider again the magnetic enthalpy (17.132), and subtract from Gvort/Lthe magnetic Gvort/L coupling hhext so that, per unit length,

1

LGh = πn2(log κ+ 0.08)− 2π

∫ ∝

0dr r h hext. (17.167)

But the integral over h is simply the flux quantum (17.144) associated with thevortex line, such that

1

LGh = πn2(log κ+ 0.08)− 2πnκ hext. (17.168)

When this quantity is smaller than zero, a vortex line invades the superconductoralong the z-axis. The associated critical magnetic field is, in natural units,

hc1 =n

2

log κ+ 0.08

κ. (17.169)

For large κ, this field can be quite small. In Fig. 17.3 we compare the asymptoticresult with a numerical solution of the differential equation for n = 1, 2, 3, . . . inFig. 17.3.

κ

hc1

n = 2

n = 1

Figure 17.3 Critical field hc1 where a vortex line of strength n begins invading a type-II

superconductor, as a function of the parameter κ. The dotted line indicates the asymptotic

result (1/2κ) log κ of Eq. (17.169). The magnetic field h is measured in natural units, which

are units of√2Hc where Hc is the magnetic field at which the magnetic energy equals the

condensation energy.


For a comparison with experiment one expresses this field in terms of the criticalmagnetic field hc = 1/

√2 and measures the ratio

Hc1

Hc

=n√2κ

(log κ + 0.08). (17.170)

If the magnetic field is increased above Hc1, more and more flux tubes invade thetype-II superconductor. These turn out to repel each other. The repulsive energyenergy between them can be minimized if the flux tubes form a hexagonal array, asshown in Fig. 17.4. The tubes will perform thermal fluctuations, which may be soviolent that the superflow experiences dissipation. The taming of these fluctuationsis one of the main problems in high-temperature superconductors. It is usually doneby introducing lattice defects at which the vortex lines are pinned.

Figure 17.4 Spatial distribution magnetization of the order parameter ρ(x) in a typical

mixed state in which the vortex lines form a hexagonal lattice [3].

17.8 Critical Field Hc2 where Superconductivity is De-stroyed

In the study of planar z-dependent field configurations we found that, for H > Hc,the order parameter vanishes, so that the magnetic field pervades the superconduc-tor. Experimentally this is not quite true. The symmetric field inside the samplecan increase markedly only up to a certain larger field value Hc2 which, however,may lie far above Hc in a deep type-II superconductor. In fact, by allowing a moregeneral space dependence, a non-zero field ρ can exist up to a magnetic field

Hc2 = κ√2Hc > Hc for κ > 1/

√2. (17.171)

Only for H > Hc2 the field ρ(x) is forced to be zero in the entire system, which thenbehaves like a normal conductor.

In order to calculate the size of Hc2, we observe that, for H very close to Hc2,the order parameter must be very small. Hence we can linearize the field equation(17.82), writing it approximately as

(i∇+ a)2 ψ ≈ ψ. (17.172)

17.8 Critical Field Hc2 where Superconductivity is Destroyed 1103

For a uniform magnetic field in the z-direction, we may choose different forms ofthe vector potential a which differ by gauge transformations. A convenient form is

a =h

κ(0, x, 0) . (17.173)

Then ψ satisfies the Schrodinger equation

−∂2x +(

i∂y +h

κx

)2

− ∂2z

ψ = ψ. (17.174)

This may be solved by a general ansatz

ψ(x, y, z) = eikzz+ikyyχ(x), (17.175)

where χ(x) satisfies the differential equation

− d2

dx2+

(

−ky +h

κx

)2

−(

1− k2z)

χ(x) = 0. (17.176)

This is the Schrodinger equation of a linear oscillator of frequency ω = h/κ, withthe potential centered at

x0 =κkyh. (17.177)

In fact, Eq. (17.176) can be written as

−χxx(x) + ω2 (x− x0)2 χ(x) = (1− kz

2)χ(x). (17.178)

This equation has a solution χ(x), which goes to zero for x→ ∞, if

1

2

(

1− k2z)

≡ En = ω(

n+1

2

)

=h

κ

(

n+1

2

)

. (17.179)

The energy En = 12(1 − k2z) is bounded by 1 from above. Hence, there cannot be

solutions for arbitrarily large h. The highest h is supported by the ground statesolution, where n = 0. But also this ceases to exist as h reaches the critical valuehc2 given by

ω

2=

h

2κ=

1

2, (17.180)

where

hc2 = κ. (17.181)

Going back to physical magnetic fields, this amounts to

Hc2 = κ√2Hc. (17.182)


The wave functions of these solutions are strongly concentrated around x ≈ x0,implying that the system is normal almost everywhere, except for a sheet of thickness√

κ/h, in units of the coherence length.

This sheet carries a supercurrent. Let us study its properties. If we insert ψ(x)of Eq. (17.175) and a of Eq. (17.173) into Eq. (17.84), we find a supercurrent density

j =1

2iψ∗ ↔

∇ ψ − |ψ|2 a = |χ(x)|2[

(0, ky, kz)−h

κ(0, x, 0)

]

= −|χ(x)|2(

0,h

κ(x− x0), kz

)

. (17.183)

The superconducting sheet is centered around the plane with x = x0 parallel to they z-plane. It carries no net charge flow along the x-axis. It consists of a double layerof current flowing in ±y directions for x > x0 or x < x0, respectively.

If the current along the z-axis is nonzero, the critical magnetic field is reducedto a lower value

hc2 → κ(

1− k2z)

. (17.184)

The sheet configuration is obviously degenerate with respect to the position x0of the sheet. Moreover, being in the regime of a linear Schrodinger equation, we canuse the superposition principle to set up different spatial structures. For example,we may form an average over sheets in the rotated (cosϕx+ sinϕy, z)-planes, andobtain cylindrical structures. These have a definite angular momentum around thez-axis rather than a definite linear momentum ky. Their wave function can be founddirectly from a vector potential

a =h

2κ(−y, x, 0), (17.185)

which reads in cylindrical coordinates

aϕ =h

2κρ, ar = 0, az = 0. (17.186)

The linearized field equation now becomes[

−(

1

r

∂

∂rr∂

∂r+

∂2

∂z2+

1

r2∂2

∂ϕ2

)

+h2

κ2r2]

= ψ(z, r, ϕ). (17.187)

It may be solved by a factorized ansatz

ψ(zrϕ) = χ(√

h/κr)eimϕeikzz. (17.188)

The differential operator in (17.187) is the same as for a two-dimensional harmonicoscillator of frequency ω = h/κ, and the solutions are well known. They may beexpressed in terms of the confluent hypergeometric functions 1F1(a; b; z) as

χ(r′) = e−r′/2r′|m|/21F1(−n; |m|+ 1; r′), (17.189)

17.8 Critical Field Hc2 where Superconductivity is Destroyed 1105

where r′ =√

h/κr. The corresponding energy eigenvalues are

E = ω(

n+1

2|m| − 1

2m+

1

2

)

+ k2z . (17.190)

They have to be equated with 1/2(1 − k2z) to obtain the highest possible magneticfield, which is again found to be

hc2 = κ(1− k2z), (17.191)

as in (17.184). The current carried by this cylindrical solution is found by inserting(17.185) and (17.188) into (17.83). For n = 0 and arbitrary m > 0, we obtain

j = χ2(

√

h/κr)

[

m(

y

r2,− x

r2, 0)

+ (0, 0, kz)−h

2κ(−y, x, 0)

]

= χ2(

√

h/κr)

[

kzez +

(

m

r2− h

2κ

)

eϕ

]

. (17.192)

The wave function is non-zero within a narrow tube of diameter√

κ/h in unitsof the coherence length. Within this tube, a current is rotating clockwise aroundthe z-axis, just as deduced before in the qualitative discussion of Eq. (17.150). Inaddition, there may be an arbitrary linear current that flows into the z-direction.

The magnetic enthalpy (17.132) of a type-II superconductor is always smallerthan zero because of the negative interfacial energy. Thus, for H below Hc2, therewill be states with nonzero magnetic field.

If the magnetic field is far below Hc2, the magnitude ρ(x) of the order field willbe so large that the quartic terms in the field equation (17.88) can no longer beneglected. The field configuration becomes inaccessible to simple analytic calcula-tions.

The different critical magnetic fields for various superconducting materials arelisted in Table 17.1.

Table 17.1 Different critical magnetic fields, in units of gauss, for various superconduct-

ing materials with different impurities.

material Hc Hc1 Hc2 Tc/K

Pb 550 550 550 4.20.85 Pb, 0.15 Ir 650 250 3040 4.20.75 Pb, 0.25 In 570 200 3500 4.20.70 Pb, 0.30 Tl 430 145 2920 4.20.976 Pb, 0.024 Hg 580 340 1460 4.20.912 Pb , 0.088 Bi 675 245 3250 4.2

Nb 1608 1300 2680 4.20.5 Nb, 0.5 Ta 252 — 1370 5.6


17.9 Order of Superconductive Phase Transition

The Ginzburg-Landau parameter κ does not only distinguish type I from type IIsuperconductors. Its magnitude distinguishes also the order of the superconductivephase transition. If κ is very large, the effect of the electromagnetic field is extremelyweak and the superconductor behaves very similar to a pure superfluid. Then thetransition is certainly of second order. In the opposite limit of small κ, the massacquired by the electromagnetic field is quite large and the penetration depth isvery small. Then the transition becomes of first order. Somewhere between theselimits, the order must change and it is possible to show that this happens at atricritical value of κ ≈ κt ≈ 0.81/

√2 [4, 5]. The theoretical tool for this was

the development of a disorder field theory [6, 7]. In it, the relevant elementaryexcitations are the vortex lines whose grand-canonical ensemble is described by afluctuating field. When vortex lines proliferate, the disorder field acquires a nonzeroexpectation value. Depending on the strength of the electromagnetic coupling e,the phase transition can be at the boundary of a first- and a second-oder phasetransition, marking a tricritical point.

17.10 Quartic Interaction and Tricritical Point

According to the action (17.9), the Ginzburg-Landau theory of superconductivity ischaracterized by the following free energy density:

f(ϕ,∇ϕ,A,∇A) =1

2(∇+ iqA)ϕ∗ (∇− iqA)ϕ

+m2

2|ϕ|2 + g

4|ϕ|4 (17.193)

+1

2(∇×A)2 ,

with the order fieldϕ(x) = ρ(x)e−iγ(x), (17.194)

where ρ(x) and γ(x) are real variables. The electromagnetic field is represented bythe vector A, and q = 2e/hc accounts for the electromagnetic coupling of Cooperpairs whose charge is 2e.3 The real constants m2 and g parametrize the size of thequadratic and quartic terms, respectively. If the mass parameter m2 drops belowzero, the ground state of the potential

V (ϕ) =1

2m2|ϕ|2 + g

4|ϕ|4 (17.195)

is obtained by an infinite number of degenerate states that satisfy

〈ϕ〉2 = ρ20 = −m2

g. (17.196)

3The Euler number is represented by the roman letter e, not to be confused with the electriccharge e.

17.10 Quartic Interaction and Tricritical Point 1107

The phase transition is of second order.If we choose the ground state to have a real field ϕ, i.e., if we choose γ(x) = 0

in (17.194), the free energy density becomes

f(ϕ,∇ϕ,A,∇A) =1

2(∇ρ)2 + V (ρ) +

q2ρ2

2A2

+1

2(∇×A)2 . (17.197)

At the potential minimum, the vector field has a mass

mA = qρ0. (17.198)

This can be observed experimentally as a London penetration depth λL = 1/mA =1/qρ0 of the magnetic field into the superconductor.

The ratio of the two characteristic length scales of a superconductor, the coher-ence length ξ = 1/

√−2m2 and the penetration depth λL, constitutes a dimensionless

material parameter of a superconductor, the so-called Ginzburg-Landau parameter

κ ≡ λL/√2ξ. Using (17.196) we find for κ the value

κ =

√

g

q2. (17.199)

A first-order phase transition arises in the Ginzburg-Landau theory by includ-ing the effects of quantum corrections, which in the mean-field approximation [8]neglecting fluctuations in ρ, lead to an additional cubic term in the potential:

Figure 17.5 Effective potential for the order parameter ρ with a fluctuation-generated

cubic term. For m2 = m2t , there exist two minima of equal height V (ρ), one at ρ = 0 and

another one around ρt, causing a first-order transition at m2 = m2t . When m2 is lowered

down to zero, the field has huge fluctuations around the origin without a stabilizing mass,

and the fluctuations move to some larger ρ = ρf > ρt with a fluctuation-generated mass

m2f .


V (ρ) =1

2m2ρ2 +

g

4ρ4 − c

3ρ3 , c =

q3

2π. (17.200)

We can see in Fig. 17.5 that the cubic term generates for m2 < c2/4g a secondminimum at

ρ2 =c

2g

1 +

√

1− 4m2g

c2

. (17.201)

At the specific point m2t = 2c2/9g, the minimum lies at the same level as the origin.

This happens at ρt = 2c/3g. The jump from ρ = 0 to ρt is a phase transition offirst-order (tricritical point). Therefore, in this point, the coherence length of theρ-field fluctuations becomes

ξt =1

√

m2 + 3gρ2t − 2cρt=

3

c

√

g

2. (17.202)

This is the same as for the fluctuations around ρ = 0.The value of the Ginzburg-Landau parameter at the tricritical point is estimated

with the help of a duality transformation to lie close to the place where type I goesover into type II superconductivity at [4, 5, 6]

κ ≈ κt =3√3

2π

√

1− 4

9

(

π

3

)

≈ 0.8√2. (17.203)

If the mass term drops to zero, the second minimum, at which the fluctuationsstabilize, has a curvature that determines a square mass

m2f = 3g, (17.204)

which corresponds to a coherence length ξf = 1/3g.

17.11 Four-Dimensional Version

Let us see how this result changes in the four-dimensional version of the Ginzburg-Landau theory, the Coleman-Weinberg model. Setting again ϕ(x) = ρ(x)e−iγ(x)

as in (17.194) (see also the O(N)-model in Chapter 22 for N = 2), the effectivepotential of the Coleman-Weinberg model at one-loop level is [9]

V (ρ) =1

2m2ρ2 +

g

4ρ4 +

3q4

64π2ρ4(

logρ2

µ2− 25

6

)

, (17.205)

where the magnitude of the scalar (spin-0) field is represented by ρ(x) =√Φ2.

Here g gives the strength of the quartic term, and µ is the value of ρ at which therenormalization is done. We shall assume g to be of the same order as q4, which isvery small. For this reason we can neglect, in the one-loop approximation (17.205),the purely scalar higher-loop corrections since these are of the order g2, which would

17.11 Four-Dimensional Version 1109

be extremely small ∝ q8. To see this clearly we define a new scale parameter M bysetting

g

4=

3q4

64π2

(

logµ2

M2+

11

3

)

, (17.206)

which turns the effective potential into

V (ρ) =1

2m2ρ2 +

3q4

64π2ρ4(

logρ2

M2− 1

2

)

. (17.207)

If the theory is massless, i.e., if m2 = 0, the potential (17.207) has a minimum atthe field value

ρ = ρf ≡M2. (17.208)

There the potential has a nonzero curvature (see Fig. 17.6).

Figure 17.6 Effective potential for the order parameter ρ in four spacetime dimensions

according to Eq. (17.205). As in three dimensions, a new second minimum develops around

ρt causing a first-order transition for m2 = m2t . For m = 0, the effective potential has a

minimum at some ρ = ρf > ρt with a fluctuation-generated mass term m2f .

The curvature implies a nonzero mass generated by fluctuations:

m2f = m2(ρf) =

∂2V

∂ρ2

∣

∣

∣

∣

∣

m=0,ρ=ρc

=q4ρ2f16π2

(

9 logρ2fM2

+ 6

)

=6q4M2

16π2. (17.209)

As described in [9, 10], the effective potential has, for a positivem2, both a maximumand a minimum as long as m2 < 2m2

1/e1/2 ≡ 3q4M2e−1/16π2. The minimum of the

potential lies at the same level as the origin if m2 = m2t ≡ 3q4M2e−1/2/32π2 with

ρ2t =M2e−1/2. This is the tricritical point, and the scalar mass at that point is

m2(ρt) =∂2V

∂ρ2

∣

∣

∣

∣

∣

ρ=ρt

= m2t +

q4ρ2t16π2

(

9 logρ2tM2

+ 6

)

=3q4M2

16π2e1/2

=λ

αρ2t , (17.210)


where

α =

(

logµ2

M2+

11

3

)

(17.211)

parametrizes the fluctuation scale M . A Ginzburg-Landau parameter may be de-fined for the four-dimensional theory, as in three dimensions, by the ratio of the twocharacteristic length scales of the theory [compare (17.199)]

κ =λL√2ξ

=1√2

m(ρt)

qρt=

1√2α

√

g

q2. (17.212)

The result has the same form as the previously obtained 3-dimensional result in(17.199). It becomes exactly the same with an appropriate choice (17.211) of therenormalization scale M .

The above U(1)-gauge-invariant framework is still far from the full description ofthe electroweak interaction, where the true symmetry group is SU(2)L×U(1)Y (seeChapter 27 and Ref. [12]).

17.12 Spontaneous Mass Generation in a Massless Theory

The most interesting property of the Coleman-Weinberg model is that it illustrateshow fluctuations are capable of spontaneously converting a classically massless the-ory into a massive theory. This is best seen by looking at the set of effective classicalpotentials for various mass terms m2ρ2/2, shown in Fig. 17.6. If m2 = 0, the curva-ture of the quadratic term in the field ∂V (ρ)/∂ρ2 at ρ = 0 is zero and the theoryis massless. The effect is that the field fluctuations around ρ = 0 diverge and makethe theory critical. The interactions of the field limit the fluctuations so that thefield expectation settles at some value ρf = M > ρt with a finite mass mf . FromEq. (17.210) we see that

m2f =

3q4M2

8π2. (17.213)

Remember that after the ρ-field acquires a nonzero expectation, the vector field hasa nonzero mass mA = qM given by Eq. (17.198). Inserting this into (17.213), weobtain the famous experimentally observable mass ratio caused by fluctuations of amassless theory:

m2f

m2A

=3

2π

q2

4π. (17.214)

The value of the effective potential at this new minimum is, according toEq. (17.207),

V (ρf ) = −3q4M4

128π2= − 3m4

A

128π2. (17.215)

Notes and References 1111

This is the important lesson of the Coleman-Weinberg model. Even though theclassical theory has a scalar field of zero mass, the fluctuations produce a nonzeromass mf via the nonzero field expectation value ρf =M . The mass mf is a so-calledspontaneously generated mass. The origin of the mass lies in the need to introducesome nonzero mass scale µ when calculating divergent loop diagrams. The newfinite mass parameterM is also referred to as the dimensionally transmuted coupling

constant of the massless theory.

Notes and References

For more information on vortex lines see the textbooksD. Saint-James, G. Sarma, E.J. Thomas, Type II Superconductivity, Pergamon, New York (1969);M. Tinkham, Introduction to Superconductivity, Dover, London, 1995.

The individual citations refer to:

[1] V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950). Engl. transl. in L.D.Landau, Collected papers, Pergamon, Oxford, 1965, p. 546.

[2] W. Meissner and R. Ochsenfeld, Naturwissenschaften 21, 787 (1933).

[3] See W.M. Kleiner, L.M. Roth, S.H. Autler, Phys. Re. A 133, 1226 (1964).

[4] H. Kleinert, Lett. Nuovo Cimento 35, 405 (1982) (http://klnrt.de/97).

[5] H. Kleinert, Europhys. Letters 74, 889 (2006) (http://klnrt.de/360).

[6] H. Kleinert, Gauge Fields in Condensed Matter , Vol. I, World Scientific, 1989(http://klnrt.de/b1).

[7] H. Kleinert, Multivalued Fields in Condensed Matter, Electromagnetism, and Gravitation,World Scientific, Singapore 2009, (http://klnrt.de/b11). See Section 5.4.3.

[8] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Finan-

cial Markets, fifth extended edition, World Scientific, Singapore 2009.

[9] E.J. Weinberg, Radiative Corrections as the Origin of Spontaneous Symmetry Breaking,Harvard University, Cambridge, Massachusetts, 1973.

[10] S. Coleman and E.J. Weinberg, Phys. Rev. B 7, 1888 (1973).

[11] H. Kleinert, Phys. Lett. B 128 69, (1983) (http://klnrt.de/106).

[12] M. Fiolhais and H. Kleinert, Physics Letters A 377, 2195 (2013) (http://klnrt.de/402).

[13] S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967).

[14] K. Nakamura et al. (Particle Data Group), J. Phys. G 37, 075021 (2010).

Documents

ScalarQuantumElectrodynamics - Freie Universitätusers.physik.fu-berlin.de/~kleinert/b6/psfiles/Chapter-16-scalarql.pdf · ScalarQuantumElectrodynamics ... In that form it is known