Scalar Quantum Electrodynamics

A complex system that works is invariably found to have evolved from a simple system that works. John Gall

In nature, there exist scalar particles which are charged and are therefore coupled to the electromagnetic ﬁeld. In three spatial dimensions, an important nonrelativistic example is provided by superconductors. The phenomenon of zero resistance at low temperature can be explained by the formation of so-called Cooper pairs of electrons of opposite momentum and spin. These behave like bosons of spin zero and charge q =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 theory to 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 theory containing a complex scalar ﬁeld

ϕ(x)= ϕ1(x)+ iϕ2(x) (17.1) coupled to a magnetic vector potential A. The scalar ﬁeld describes bound states of pairs of electrons, which arise in a superconductor at low temperatures due to an attraction coming from elastic forces. The detailed mechanism will not be of interest here; we only note that the pairs are bound in an s-wave and a spin singlet state of charge q =2e. Ignoring for a moment the magnetic interactions, the ensemble of these bound states may be described, in the neighborhood of the superconductive 4 transition temperature Tc, by a complex scalar ﬁeld theory of the ϕ -type, by a euclidean action

2 3 1 m g 2 E = d x ∇ϕ∗∇ϕ + ϕ∗ϕ + (ϕ∗ϕ) . (17.2) A Z " 2 2 4 # The mass parameter depends on the temperature like m2 (T T )/T . Above ∝ − c c the 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 discussed in Chapter 16. The symmetry group that is broken, is given by the U(1) phase transformation

iα ϕ(x) e− ϕ(x) (17.3) → under which is obviously invariant. Alternatively we may write AE ϕ cos αϕ + sin αϕ , 1 → 1 2 ϕ sin αϕ + cos αϕ , (17.4) 2 → − 1 2 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 in D = 2+1 dimensions, with an action

2 D 1 µ m g 2 E = d x ∂µϕ∗∂ ϕ ϕ∗ϕ (ϕ∗ϕ) . (17.5) A Z "2 − 2 − 4 # For the sake of generality, we shall discuss this theory for arbitrary D. Most formulas will be written down explicitly in D = 4 dimensions, to emphasize analogies with proper QED discussed in Chapter 12. The phenomenon of spontaneous symmetry breakdown in this system has been studied in detail in Chapter 16. For T > Tc, where the mass term is positive, small oscillations around ϕ = 0 consist of two degenerate modes carried by ϕ1,ϕ2, both 2 of mass m . For T < Tc, where the mass is negative, the energy is minimized by a ﬁeld ϕ with a real non-vanishing vacuum expectation value, say ϕ(x) ϕ . (17.6) h i ≡ 0 Then the symmetry between ϕ1 and ϕ2 is broken and there are two diﬀerent modes of small oscillations: one orthogonal to ϕ , which is the massless Nambu-Goldstone mode, and one parallel to ϕ , which hash i a positive mass 2m2 > 0. h i − In order to describe the phenomena of superconductivity with the euclidean version of (17.5), we must include electromagnetism. According to the minimal substitution rules described in Chapter 12, we simply replace ∂µϕ in (17.5) by the covariant derivative ∂ ϕ D ϕ (∂ iqA ) ϕ. (17.7) µ → µ ≡ µ − µ The extra vector potential Aµ(x) is assumed to be governed by Maxwell’s electromagnetic action [see (12.3)]:1

1 D µν γ = d xFµν (x)F (x). (17.8) A −4 Z The combined action becomes 2 D 1 2 m 2 g 4 1 2 se[ϕ,ϕ∗, A]= d x Dµϕ ϕ ϕ Fµν . (17.9) A Z (2| | − 2 | | − 4| | − 4 ) 1In this chapter we use natural units with ¯h = c = 1. 17.1 Action and Generating Functional 1079

2 µν where Fµν is short for FµνF . This action is invariant under local gauge transformations ϕ(x) eiqΛ(x)ϕ(x), A (x) A (x)+ ∂ Λ(x). (17.10) → µ → µ µ A functional integration leads to the generating functional of scalar QED:

∗ µ ∗ ∗ µ µ i se[ϕ,ϕ ,A ] λ ϕ ϕ λ j Aµ Z [λ,λ∗, j] = ϕ ϕ∗ A e {A − x − x − x }, (17.11) D D phys. D Z Z R R R 4 where the symbol x abbreviates the volume integral d x, and λ(x), λ∗(x) are the µ sources of the complexR scalar fields ϕ∗(x) and ϕ(x). TheR vector j (x) is an external electromagnetic current coupled to the vector gauge field Aµ(x). To preserve the gauge invariance, the current is assumed to be a conserved quantity and satisfy ∂ jµ(x) = 0. The functional integration over the vector field Aµ assumes µ phys D that a Faddeev-Popov gauge fixing factor is inserted, together withR a compensating determinant (recall the general discussion in Section 14.16):

µ µ 1 A A Φ− [A]F [A]. (17.12) Zphys D ≡ Z D For m2 > 0, we may choose [recall (14.353)]

1 µ 2 (∂µA ) F [A]= F4[A]= e− 2α x , (17.13) R and derive from (17.11) the Feynman rules for perturbation expansions. The gauge- ﬁxed photon propagator was given in Eq. (14.373):

D d k ik(x y) i kµkν − − (17.14) G0 µν(x y)= D e 2 gµν (1 α) 2 . − − Z (2π) k " − − k # We have seen in Chapter 12 that in QED all scattering amplitudes are independent of the gauge parameter α. In addition, the perturbation expansion derived from the generating functional (17.11) employs propagators of scalar particles represented by lines

D ik(x y) d k ik(x y) i e− − G (x y)= e− − 0 − (2π)D k2 m2 (17.15) Z − and vertices

. (17.16)

Since the scalar ﬁeld is complex, the particle carries a charge, and the charged particle lines in the Feynman diagrams carry an orientation. The coupling of the scalar ﬁeld to the vector potential does not only give rise to a three-point vertex of the type (12.95) in QED [left-hand diagram in (17.16)], but also to a four-vertex in which two photons are absorbed or emitted by a scalar particle [right-hand diagram in (17.16)]. This diagram is commonly referred to as seagull diagram. 1080 17 Scalar Quantum Electrodynamics

17.2 Meissner-Ochsenfeld-Higgs Eﬀect

2 The situation is drastically diﬀerent for T < Tc where m < 0. At the classical level, the lowest energy state of the pure ϕ theory is reached at ϕ = ϕ0 with ϕ = φ = m2/g, i.e., | 0| 0 − q 2 iγ0 m iγ0 ϕ0 = φ0e− = − e− . (17.17) s g Now the photon part of the action (17.9) reads

2 2 D 1 2 q m 2 γ = d x Fµν Aµ . (17.18) A Z −4 − 2 g ! Extremization of this leads to the Euler-Lagrange equation

q2m2 ∂2A ∂ ∂A A =0. (17.19) µ − µ − g µ Note that we could have chosen as well

2 iγ0(x) m iγ0(x) ϕ = φ0e− = − e− , (17.20) s q with an arbitrary spacetime-dependent phase γ0(x). Then the photon Lagrangian would read

2 2 D 1 2 q m 2 γ = d x Fµν (Aµ ∂µγ0) , (17.21) A Z "−4 − 2 g − # with the Euler-Lagrange equation

m2 ∂2A ∂ (∂A) q2 (A ∂ γ )=0. (17.22) µ − µ − g µ − µ 0 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 transformation A A + ∂ γ (x), and the physical content of (17.21) is independent of the µ → µ µ 0 particular choice of γ0(x). Consider now the ﬂuctuation properties of this theory. The generating functional 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 into radial and azimuthal parts

iγ(x) ϕ(x)= ρ(x)e− , (17.24) 17.2 Meissner-Ochsenfeld-Higgs Eﬀect 1081 the path integral takes the form

∗ µ ∗ ∗ µ µ i se[ϕ,ϕ ,A ] λ ϕ ϕ λ j Aµ Z [λ,λ∗, j]= ρ ρ γ A e {A − x − x − x }. (17.25) D D phys D Z Z R R R But from the structure of the covariant derivative it is obvious that no matter what the fluctuating γ(x) configuration is, it can be absorbed into A(x) as a longitudinal 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 transverse and longitudinal Aµ(x)-fluctuations. This may be expressed by rewriting the functional integral (17.25) as

µ ∗ ∗ µ µ i se[ρ,A ] λ ϕ ϕ λ j Aµ Z [λ,λ∗, j]= ρ ρ A e {A − x − x − x }, (17.26) D all D Z Z R R R with the action

2 2 µ D 1 2 q 2 2 m 2 g 4 1 2 [ρ, A ]= d x (∂µρ) + ρ Aµ ρ ρ + Fµν . (17.27) A Z "2 2 − 2 − 4 4 # Due to the second term in the exponent, the formerly irrelevant gauge degree of freedom 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- 1 tional F [A] of (17.13) and a compensating Faddeev-Popov determinant Φ− [A] of Eq. (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 with each other by a gauge transformation which involves the two fields (or any other field 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[A ,ϕ]= Λ F AΛ,ϕΛ . (17.29) µ D Z h i It is most conveneiant to choose a gauge condition which ensures the reality of the complex scalar ﬁeld ϕ via the δ-functional:

F [Aµ,ϕ]= δ[ϕ2(x)], (17.30) where ϕ2(x) = Im ϕ(x). Since the gauge transformations change

iΛ(x) ϕ e− ϕ = [cosΛ(x) ϕ + sin Λ(x) ϕ ]+ i [ sin Λ(x) ϕ + cosΛ(x)ϕ ] , (17.31) → 1 2 − 1 2 we see that

Φ[Aµ,ϕ]= Λ δ [ sin Λ ϕ1 + cosΛ ϕ2] . (17.32) Z D − 1082 17 Scalar Quantum Electrodynamics

The result must be independent of the choice of gauge and can therefore be obtained at ϕ =( ϕ , 0), where we ﬁnd | | 1 Φ[A ,ϕ]= . (17.33) µ ϕ(x) | | Note that Φ involves no derivative terms as in the earlier gauges. For this reason iγ(x) it does not require Faddeev-Popov ghost ﬁelds. Setting ϕ = ρ(x)e− , we arrive precisely 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 form iγ expected from the radial-azimuthal decomposition ϕ = ρe− , and the associated integration measure dϕdϕ∗ = dϕ1dϕ4 = ρdρdγ. The generating functional (17.26) may now be used for perturbation expansions below Tc. Inserting into (17.27)

m2 ρ(x)= ρ0 + ρ′(x)= + ρ′(x), (17.34) s− g we can rewrite 2 D 1 2 ( 2m ) 2 g 4 3 3 = d x (∂µρ′) − ρ′ ρ′ gρ0ρ′ A Z " 2 − 2 − 4 − 2 2 2 1 2 µ 2 2 q 2 2 Fµν + Aµ + q ρ0ρ′Aµ + ρ′ Aµ , (17.35) −4 2 2 # with m2 µ2 q2 . (17.36) ≡ − g This is the mass by which the Nambu-Goldstone theorem is violated. The model exhibits a spontaneous breakdown of the continuos U(1)-symmetry which by Noether’s theorem should produce a massless excitation. But the mixing of this excitation with the 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” the would-be Nambu-Goldstone boson and grown “fat”. The same mechanism has been used to unify the theories of electromagnetic and weak interactions. Then the scalar field ϕ possesses more components than the real and imaginary parts of the complex field ϕ, and the associated Lagrangian has a higher symmetry than U(1). In that case the analog to the ρ-mass 2m2 m2 = − (17.37) ρ 2 is referred to as the Higgs mass [see Eq. (27.99)], and the mass (17.36) of the fat vector field is referred to as the vector boson mass. This will be discussed in detail in Chapter 27. 17.2 Meissner-Ochsenfeld-Higgs Effect 1083

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

dq iqx i ρ′(x)ρ′(0) = e− . (17.38) (2π)4 q2 ( 2m2)+ iη Z − − 3 4 The ρ′-ﬁeld has cubic ρ′ - and quadratic ρ′ -interactions. The originally massless Goldstone ﬁeld γ has disappeared. It has been “eaten up” by the photon, making it massive, and giving it one more polarization degree of freedom. In the action (17.35) there are electromagnetic vertices in which one or two particles couple to two photons

2 q ρ0 . (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

1 dDk = A ( k) kµkν gµν k2 µ2 A (k), (17.40) A −2 (2π)4 µ − − − ν Z h i where the kinetic matrix between the ﬁelds is

M µν = kµkν gµν k2 µ2 (17.41) − − kµkν kµkν = gµν k2 µ2 + µ2 . (17.42) − − k2 ! − k2 By inverting this we ﬁnd the matrix

µν µ ν 2 µ ν µν µ ν 2 1 g k k /k 1 k k g k k /µ M − = G (k)= − + = − (17.43) µν µν − k2 µ2 µ2 k2 − k2 µ2 − − so that the massive propagator becomes

dk ikx dk ikx i kµkν Gµν(x)= e− Gµν(k)= e− − gµν . (17.44) (2π)4 (2π)4 k2 µ2 − k2 ! Z Z − 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 and can invade only a thin surface layer of penetration depth 1/µ. 2 2 At k = µ , there is a massive particle pole. Since the propagator Gµν (k) at this pole is purely transverse, it satisfies

µ k Gµν (k)=0. (17.45)

The massive particles have three internal degrees of freedom corresponding to the three orientations of their spin. This has to be contrasted with the massless gauge 0 invariant theory for T > Tc, where gauge invariance permits to choose, say, A =0 1084 17 Scalar Quantum Electrodynamics while still satisfying (17.45) at the pole, thereby eliminating one more degree of freedom in the asymptotic states. Only two transverse photon polarizations remain far away from the interaction. There is no contradiction in the number of degrees of freedom since for T > Tc there are two modes carried by the complex ϕ fields. The third photon degree is a consequence of absorbing, into the vector field, the phase oscillations, i.e., the former Nambu-Goldstone degrees of freedom. Certainly, the same theory can also be calculated with the earlier gauge fixing Lagrangian 1 = (∂µA )2 , (17.46) LGF −2α µ in which case the quadratic piece of the action is

2 quad D 1 2 ( 2m ) 2 = d x (∂µρ′) − ρ′ (17.47) A Z " 2 − 2 2 2 1 2 µ 1 1 µ 2 Fµν + Aµ ∂µγ (∂µA ) . −4 2 − q ! − 2α   1 Recalling the Faddeev-Popov determinant Φ4− of (14.365), we have to add for this gauge the Faddeev-Popov ghost action corresponding to (14.418):

1 D µ ghost = d x ∂µc∗∂ c . (17.48) A −2α Z The generating functional is obtained by integrating the exponential ∗ ∗ µ i se+ ghost λ ϕ ϕ λ j Aµ e {A A − − − } over all Aµ(x), all complex ϕ(x), and all ghost conﬁg- urations c∗(x),c(x):

∗ ∗ µ i se+ ghost λ ϕ ϕ λ j Aµ Z [λ,λ∗, j]= c c∗ ϕ ϕ∗ Aµ e {A A − − − }. (17.49) Z D D Z D D Z D What is the particle content of this theory? Since the Faddeev-Popov ghosts do not interact with the other ﬁelds in the action, they can be ignored. As far as the photon is concerned, let us split the vector potential as

t l Aµ = Aµ + Aµ, (17.50)

t µ t where A is the transverse part satisfying the Lorenz gauge condition ∂ Aµ = 0, and l l 2 ν 2 A is the longitudinal part Aµ = ∂− ∂µ∂ν A , which is a pure gauge ﬁeld. Then the transverse part has the propagator

D t t d k i kµkν Aµ(x)Aν (0) = − gµν . (17.51) (2π)D k2 µ2 − k2 ! Z − 2As in (14.370), the letters l and t stand for longitudinal and transverse to the four-vector ﬁeld Aµ(x). 17.2 Meissner-Ochsenfeld-Higgs Eﬀect 1085

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 scalar field γ′ (1/q) γ, we find the quadratic Lagrange density in energy-momentum space ≡ 2 2 1 4 2 µ k α− k k Λ (Λ,γ′) − 2 − 2 , (17.52) 2 k k ! γ′ ! − which has the determinant µ4/4αk6, and eigenvalues − µ2k2 1 k2 k2 2 λk = 1 2 v1+ 2  . (17.53) 2 − 2α µ ± u 2αµ ! u  t    Both vanish at k2 = 0 (and only there). Thus there are two more massless asymptotic states in Hilbert space. At first sight, this formulation describes a physical situation different from the previous one. We do know, however, that the generating functionals are the same. In the previous formulation, there were only two types of particles. Therefore all additional zero-mass asymptotic states found in the present gauge must be such that, if a scattering process takes place with the two physical particles coming in, the unphysical ones can never be produced. Phys- ical and unphysical states should have no mutual interaction. We may say that the S-matrix is irreducible on the space of physical states. For the Faddeev-Popov ghost fields this is trivial to see. With respect to the other states, however, it is not so obvious, and the path integral is the only efficient tool for convincing us of the correctness of this statement, as demonstrated in Section 14.16. The irreducibility of the S-matrix becomes most transparent by using a particular gauge due to ’t Hooft in which physical and unphysical parts of the Hilbert space receive a clear distinction which depends on the gauge parameter. Consider the gauge condition

2 2 1 µ µ F [A, ϕ] = exp ∂µA αγ . (17.54) −2α − q !    For α 0 this enforces the Lorenz gauge. The exponent has the pleasant advantage → that, 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 required 1 in (17.52). Therefore, postponing the calculation of Φ− [A, ϕ] to Eq. (17.60), the gauge-ﬁxed free action reads 2 2 free D 1 2 2m 2 1 2 µ 2 = d x (∂ρ′) − ρ′ Fµν + Aµ A Z (2 − 2 − 4 2 2 µ 2 2 2 1 µ 2 + (∂µγ) αµ γ (∂µA ) . (17.55) 2q2 − − 2α ) h i Thus the photon has a quadratic piece 1 1 Aµ( k) k2 µ2 g + k k k k Aν(k) 2 − − − µν µ ν − α µ ν 1086 17 Scalar Quantum Electrodynamics

2 2 µ ν 1 µ 2 2 kµkν k αµ k k ν = A ( k) k µ qµν − A (k), (17.56) 2 − "− − − k2 ! − α k2 # which by inversion of the matrix between the ﬁelds leads to a propagator

D µ ν d k ikx i kµkν iα k k Gµν (x)= e− gµν . (17.57) (2π)4 "−k2 µ2 − k2 ! − k2 αµ2 k2 # Z − − Apart from the physical propagator, there is a ghost-like state in Aµ(k) parallel to kµ. 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 the gauge ﬁxing term destroys explicitly also the global invariance of the Lagrangian

iΛ ϕ(x) e− ϕ(x), γ(x) γ(x)+Λ, A A , (17.58) → → µ → µ under which the gauge pieces transform as

2 2 2 1 µ µ 1 µ µ 2 ∂µA αγ ∂µA α(γ + Λ) . (17.59) −2α − q ! → −2α " − q #

Therefore, pure phase transformations change the energy, and this results in a mass term. Let us now take a look at the functional Φ[A, ϕ] in order to see that this does not modify the Lagrangian (17.55). According to (14.354), we have to integrate

1 µ2 2 Φ[A, ϕ] = Λ exp ∂µAΛ αγΛ  µ  Z D −2α − q !   1 µ2α 2 = Λ exp ∂µA + ∂2Λ (γ + Λ) (17.60)  µ  Z D −2α − q !   1 µ2 µ2 2 = Λ exp ∂µA αγ + ∂2 α Λ  µ  Z D −2α − q − q ! !   which, after a shift of the integration variable, gives the trivial constant

µ2 Φ[A, ϕ] = det ∂2 + α /√α. (17.61) − q ! This can be expressed in terms of Faddeev-Popov ghost ﬁelds as

2 1 i µ Φ− [A, ϕ]= c c∗ exp dx ∂c∗∂c + αc∗c . (17.62) Z D D "√α Z q !# The situation is similar to that of the last gauge condition: There are three additional kinds 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. But contrary to the previous case, all these states now have a mass which depends on the gauge parameter α. This proves that they must be artifacts of the gauge choice and cannot contribute to physical processes. In particular, by letting the parameter α tend to infinity, all these particles become infinitely heavy and are thus eliminated from the physical Hilbert space.

17.3 Spatially Varying Ground States

The theory of a complex scalar field interacting with a photon has another interesting feature. The ground state must not necessarily be uniform. There are certain ranges of mass and external source where the system prefers to be filled with string-like field configurations, and others where the fields have spatially periodic structures of the hexagonal type. In order to see how this comes about and to be able to compare results with experimental data we consider the euclidean three-dimensional scalar QED, 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 the elastic forces that the attraction overcomes the Coulomb repulsion. Then they form Cooper 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 action is

2 1 3 1 2 1 m 2 g 4 Γ[Φ] = d x H + (DΦ)∗ DΦ+ Φ + Φ . (17.63) T − Z "2 2 2 | | 4| | # Since the charge carriers are electron pairs, the charge of the field is twice the electronic charge, and the covariant derivative (17.7) becomes 2e D = ∇ iqA = ∇ i A. (17.64) − − hc¯ The first expression is written down in natural units withh ¯ = c = 1, the second in physical 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)2 Φ+ 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

q ↔ 2 2 j = Φ∗ ∇ Φ q Φ A. (17.68) 2i − | | 1088 17 Scalar Quantum Electrodynamics

In order to describe superconductivity in the lower neighborhood of the critical temperature, we insert a temperature dependent mass term T m2 = µ2 1 = µ2τ, (17.69) Tc − − 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 Φ ψ = , (17.70) Φ | 0| with

µ2 µ Φ0 = = √τ (17.71) | | s− g √g being the nonzero vacuum expectation of the ϕ-field for T < Tc. We also define a length scale associated with the massive fluctuations of the Φ-particle 1 ξ(τ)= . (17.72) µ√2 A second length scale charecterizes the range of the photon, after it has “eaten up” the Nambu-Goldstone boson: 1 √g √g λ(τ)= = = ξ(τ). (17.73) q Φ qµ√τ q | 0| The first length scale is usually referred to as coherence length, the second as penetration depth, for reasons which will become plausible soon. The ratio of the two length scales is an important temperature-independent material constant, called the Ginzburg-Landau parameter and denoted by κ:

λ(τ) √g κ = , (17.74) ≡ ξ(τ) q which measures the ratio between the coupling strength g with respect to q2. The coherence 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 Γ [Φ] = 4f ξ3F,¯ (17.77) T − c where fc is the so-called condensation energy density: Φ2m2 m4 µ4 f = 0 = = τ, (17.78) c 4 − 4g −4g and F¯ is the reduced free energy 1 1 1 1 1 ¯ 3 ∇ a 2 2 4 2 h2 F = 3 d x ξ(τ)( i )ψ ψ + ψ + λ (τ) . (17.79) ξ Z 2 | − | − 2| | 4| | 2 Note that h measures the magnetic ﬁeld in units of 1 H = √2H = m2Φ2 =2 f , (17.80) 0 c ≡ qλξ 0 c q q 2 where Hc is the value at which the magnetic energy density Hc /2 equals the condensation 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 1 1 1 F¯ = d3x (∇ ia) ψ 2 ψ 2 + ψ 4 + h2 . (17.81) Z 2 | − | − 2| | 4| | 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

1 ↔ 2 j = ψ∗ ∇ ψ ψ a. (17.84) 2i −| | 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 go over to the field a a ∇γ. (17.86) → − This brings F¯ to the simple form 1 1 1 F¯ = dx (∇ ia) ρ 2 ρ2 + ρ4 +(∇ a)2 , (17.87) Z 2 | − | − 2 4 × and the field equations (17.82) and (17.83) become 2 ∇2 + a2 1+ ρ2 ρ = 0, (17.88) − − κ∇ h = κ2∇ (∇ a) = κ2[∇2a ∇ (∇ a)] = ρ2a. (17.89) × × × − · − These will be most suitable for upcoming discussions. 1090 17 Scalar Quantum Electrodynamics

17.5 Planar Domain Wall

As mentioned earlier, it has been known for a long time that superconductors have the tendency of expelling magnetic fields (Meissner-Ochsenfeld effect) [2]. Let us study what the free energy (17.87) has to say about this magnetic phenomenon. In order to get some rough ideas it is useful to consider a material sample in a magnetic field that points in the x-direction. Let us allow for variations of the system only along the z-coordinate. We may choose the vector potantial a0 to point purely into the x-direction. If we denote the x-component of a(z) a (z), the fields are ≡ 0

hy(z)= κa′(z), hx = hy =0. (17.90)

The reduced free energy density reads

1 2 1 2 1 4 2 2 2 2 f¯ = ρ′ ρ + ρ + a ρ + κ a′ . (17.91) 2 − 2 4 Its extremization yields the equation of motion for ρ(z):

2 3 ρ′′ + a ρ = ρ ρ , (17.92) − − and for a(z):

2 2 κ a′′ = aρ . (17.93)

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

2 2 d 1 dh 2 2h′ρ′ 2 κ ρ = κ h′′ = hρ . (17.94) dz ρ2 dz ! − ρ !

Inserting (17.93) into (17.92) yields

2 2 h′ 3 ρ′′ + κ = ρ ρ . (17.95) − ρ3 −

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 h a = ′′ = ′ . (17.96) ρ2 ρ2

This equation gives an immediate result: A constant magnetic ﬁeld can exist only for ρ = 0, i.e., if the system has no spontaneous symmetry breakdown. If we assume, conversely, that ρ = const. = 0, then Eq. (17.92) shows that also a2 is a constant, 2 2 6 namely a = (1 ρ ) /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 consistent solution. Therefore, constant solutions have only the two alternatives

h = const =0, ρ =0, normal phase (17.97) 6 h = 0, ρ =1, superconductive phase. (17.98)

The second alternative exhibits the experimentally observed Meissner-Ochsenfeld effect [2], implying that the superconductive state does not permit the presence of a magnetic field. Consider now z-dependent field configurations. In order to simplify the discussion 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 may propagate only over length scales λ(T ) = κξ(τ) or, in natural units, z 1, i.e., it has a unit range. For very small κ, Eq. (17.95) becomes ∼

2 ρ′′ ρ 1 ρ , (17.99) ≈ − which can be integrated by multiplying it with ρ′ to ﬁnd

2 1 2 2 ρ′ 1 ρ + E. (17.100) − ≈ 2 − If the system is in the superconductive state for large z, the field satisfies the boundary 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 dρ z = √2 = √2 Atanh ρ, (17.101) 1 ρ2 Z − or z ρ(z) = tanh . (17.102) √2 The field configuration is displayed on the right-hand side of the upper part of Fig. 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. 1092 17 Scalar Quantum Electrodynamics

Otherwise, there would be energy flow. Inserting (17.100) for E = 0 into (17.91), it becomes approximately 1 1 1 fˆ + a2ρ2 + h2. (17.104) ≈ −4 2 2 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 1 h = hc = . (17.105) √2 Within the region around z 0, the magnetic field h drops to zero over a unit length ≈ scale determined by (17.93), which is very short compared with the length 1/κ over which ρ varies (this is precisely the coherence length). If we plot the transition region against z, the field h jumps abruptly to zero from h = hc while ρ has a smooth increase. The full approximate solution plotted in Fig. 17.1 is

h(z) hcΘ( z), (17.106) ≈ −z ρ(z) tanh . (17.107) ≈ √2

ρ H

κ 1, type I ≪ N S

κ 1, type II ≫

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 approximate (17.95) by

2 2 4 2 κ h′ ρ 1 ρ , (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 =1at z = . Then h must decrease for positive z and we have to take the positive square-root∞

2 2 κh′ = ρ 1 ρ . (17.110) − q From Eq. (17.94), on the other hand, we obtain

d 1 d d h κ2 h = κ 1 ρ2. (17.111) ≈ dz ρ2 dz − dz − ! q It is convenient to introduce the abbreviation

u 1 ρ2, (17.112) ≡ − q such that (17.111) becomes

h = κu′, (17.113) − and (17.110) turns into the simple diﬀerential equation

2 2 κ u′′ = 1 u u. (17.114) − After multiplying this with u′, and integrating, we ﬁnd

2 2 2 2 u κ u′ = u 1 + const. (17.115) − 2 ! 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 equation is then solved by du 1 1 z z0 = κ = κatanh (17.116) − Z u 1 u2/2 1 u2/2 − − q q or √2 u(z)= . (17.117) cosh[(z z )/κ] − 0 From (17.112) we find

2 ρ = 1 2 . (17.118) s − cosh [(z z )/κ] − 0 As z comes in from large positive values, ρ decreases. We can arrange it to become zero at z = 0 if we choose z0 such that

sinh(z0/κ)=1. (17.119) 1094 17 Scalar Quantum Electrodynamics

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

√2 sinh[(z z )/κ] h(z)= − 0 . (17.120) cosh2[(z z )/κ] − 0 It is zero at z = , and becomes gradually stronger as it approaches z = 0, where ∞ it reaches the value 1 h = hc = , (17.121) √2 as before. From there on we may again match the trivial solution continuously by setting

h h , ρ 0; z < 0. (17.122) ≡ c ≡ The full solution shown in Fig. 17.1 is then 2 ρ(z) = Θ(z) 1 2 , (17.123) s − cosh [(z z )/κ] − 0 √2 sinh[(z z )/κ] h(z) = Θ( z)h + Θ(z) − 0 . (17.124) − c cosh2[(z z )/κ] − 0 The ranges, over which h and ρ vary, are of equal order κ, or in physical units of order λ = κξ.

Case III : κ =1/√2 For this κ-value, a trial ansatz 1 h = 1 ρ2 (17.125) √2 − can be inserted into (17.94), and leads to the second-order diﬀerential equation

2 2 2 1 ρ ρ = ρρ′′ + ρ′ . (17.126) − − This, in turn, happens to coincide with the other equation (17.95), thus confirming the correctness of the ansatz (17.125). For z = , the fields start out with ρ = 1 and h =0, and go to ρ = 0, and h = h =1/√2∞ for z . c → −∞ Equation (17.126) takes a particularly appealing form if we consider the auxiliary function σ(z) defined by ρ(z) eσ(z)/2. Then ≡ 1 σ/2 1 σ/2 1 2 σ/2 ρ′ = σ′e , ρ′′ = σ′′e + σ′ e , (17.127) 2 2 k and we may rewrite (17.126) as

2 σ ρ′′ ρ′ 1 (1 e )= = σ′′. (17.128) − − ρ − ρ2 2 17.6 Surface Energy 1095

This can be integrated to σ 2 eσ 1 σ = ′ , (17.129) − − 4 which has the solution σ 1 dσ′ z = ′ . (17.130) −2 a √eσ 1 σ Z − − ′ z/√2 For z , the solution σ(z) goes to zero like e− , 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 all look qualitatively similar. The ratio κ of penetration depth to coherence length determines how far the magnetic ﬁeld invades into the superconductive region relative to the coherence length.

17.6 Surface Energy

Consider the energy per unit length for these classical fields as they follow from Eq. (17.91). Inserting the equations of motion and performing one partial integration renders the much simpler expression for the energy per area A¯, both in natural units: ¯ F 1 4 1 2 ¯ = dzf = dz ρ + h . (17.131) A Z Z −4 2 Obviously, the classical solution with an absolute minimum is h = 0, ρ = 1. In order to enforce the previously discussed configurations, an external magnetic field is needed, and we must minimize the reduced magnetic enthalpy per unit length area ¯ G 1 4 1 2 ¯ = dz ρ + h h hext , (17.132) A Z −4 2 − where we have subtracted a term

m hext = h hext, (17.133) in which the reduced magnetization m ∂hG/¯ A¯ = h is coupled to the reduced external 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 h , the enthalpy goes asymptotically against → −∞ → → c ¯ G 1 2 ¯ dz hc hc hext . (17.134) A ≈ Z 2 − For z , on the other hand, where the size of the order field goes to 1 and h 0, the asymptotic→∞ value is → G¯ 1 ¯ dz . (17.135) A ≈ Z −4 1096 17 Scalar Quantum Electrodynamics

Thus both asymptotic regimes have the same magnetic enthalpy for the particular external 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 that side, 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 is uniformly super-conductive. We now calculate the energy stored in the finite region around z = 0 at the critical magnetic field hext = hc. There the density deviates only slightly from the asymptotic form, and we must evaluate the expression

G¯ G¯ 1 1 1 1 − as = ∞ dz ρ4 + h2 h + . (17.136) A¯ −4 2 − √2 4! Z−∞ 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¯ G¯ u2 1 u2 1/2 − as = ∞ dz u2 1 u 1 A¯  − 2 ! − √2 − 2 !  Z−∞   u2 1 du 2 1 = ∞ dz u 1 = √2 < 0. (17.137)  s − 2 − √2 dz −3 − 2 Z−∞   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 = hc by Eq. (17.105), we ﬁnd

G¯ G¯ 0 1 1 1 − as = dz 1 ρ4 = dρ (1 + ρ2), (17.138) A¯ 4 − 2√2 0 Z−∞ Z 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 hard type-I superconductor.

17.7 Single Vortex Line and Critical Field Hc1

Consider now a type-II superconductor in a small external magnetic field Hext, where it is in the state of a broken symmetry with a uniform order field Φ = Φ0. When increasing Hext, there will be a critical value Hc1 where the field lines first begin to invade 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 will perforate the superconductor, until the critical magnetic field Hc is reached, where the sample becomes normally conducting. The regime between Hc1 and Hc is called the mixed state of the type-II superconductor. The quantum of flux carried by each flux tube is

ch 7 2 Φ = 2 10− gauss cm . (17.139) 0 2e ≈ × 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 superfluid helium discussed in Section 16.4. The two objects possess, however, quite different physical properties, as we shall see. Suppose the system is in the superconductive state without an external voltage so that there is no net-current j. Suppose a flux tube runs along the z-axis. Then we can use the current (17.84) to find the vector potential

j 1 1 ↔ a = + ψ† ∇ ψ. (17.140) − ψ 2 2i ψ 2 | | | | Far away from the ﬂux tube, the state is undisturbed, the current j vanishes, and we have the relation

1 1 ↔ a = ψ† ∇ ψ. (17.141) 2i ψ 2 | | 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 in Section 16.4. There the superﬂow velocity was proportional to the gradient of the phase 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 applies here:

dγ(x)= dx ∇γ(x)=2πn. (17.143) IB IB · By Stokes’ theorem, this is equal to the integral dS ∇ a, where dS is a surface · ×∇ element of the area enclosed by the circuit. SinceR h = κ( a) is the magnetic ﬁeld in natural units [recall (17.86)], the integral (17.143) is directly× proportional to the magnetic ﬂux through the area of the circuit

¨¯ = dS h = κ dS (∇ a)= κ dx a = κ dx ∇γ =2πnκ. (17.144) ZSB · Z · × IB · IB · 1098 17 Scalar Quantum Electrodynamics

This holds in natural units, indicated by a bar on top of ¨, in whichh ¯ = c = 1. The vector potential is given by (17.75), and x is measured in units of the coherence length. 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 2e q = (17.145) hc¯ if Eq. (17.64) is expressed in CGS-units, the relation (17.141) reads

2e 1 ↔ A = ψ† ∇ ψ, (17.146) 2ihc¯ ψ 2 | | and (17.142) becomes

hc¯ ¨ A(x)= ∇ γ(x)= 0 ∇ γ(x). (17.147) 2e i 2π i The magnetic ﬂux integral in CGS-units is therefore

Φ0 ¨ ¨ = dS H = dS (∇ A)= dx A = dx ∇γ = n 0, (17.148) ZSB · Z · × IB · 2π IB · 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. The angular integral dx ∇γ still has to be equal to 2πn, and we find the quantization · rule H j dx A + =2πn, (17.149) B · ψ 2 ! I | | or ¯ 1 ¨ + dx j =2πnκ. (17.150) ψ 2 B · | | I This shows that a smaller circuit, which encloses fewer magnetic field lines, contains an 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 shows that the flux tube is also a vortex line with respect to the supercurrent. The circular current density must be inversely proportional to the distance. This behavior will be derived explicitly in Eq. (17.192). Quantitatively, we can find the properties of a vortex line by solving the field equations (17.88) and (17.89) in cylindrical coordinates. Inserting the second into the first equation, we find

1 d dρ κ2 dh 2 r + (1 ρ2)ρ =0, (17.151) −r dr dr ρ3 dr ! − − 17.7 Single Vortex Line and Critical Field Hc1 1099 where h denotes the z-component of h. Forming the curl of the second equation gives the cylindrical analogue of (17.96), i.e.,

1 d r d h = κ2 h. (17.152) r dr ρ2 dr

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

h′(r)=0, r . (17.153) →∞ In cylindrical coordinates, the ﬂux quantization condition (17.144) can be written in the form

∝

¨¯ =2π drrh(r)=2πnκ. (17.154) Z0 Inserting here Eq. (17.152) yields

∞ 2 r 2 r

¨¯ ′ ′ =2πκ 2 h (r) = 2πκ 2 h (r) , (17.155) "ρ #0 − π r=0 so that the quantization condition turns into a boundary condition at the origin:

2 n 1 h′ ρ , for r 0. (17.156) → − κ r → Inserting this condition into (17.151) we see that, close to the origin, ρ(r) satisﬁes the equation

1 d d n2 r ρ(r)+ ρ (1 ρ2)ρ 0, (17.157) −r dr dr r2 − − ≈ implying the small-r behavior of ρ(r): r n ρ(r)= c + rn+1 . (17.158) n κ O Reinserting this back into (17.156) we have

c2 r 2n h(r)= h(0) n . (17.159) − 2κ κ For large r, where ρ 1, the diﬀerential equation (17.152) is solved by the modiﬁed → Bessel function K0, with some proportionality factor α: r h(r) αK0 , r . (17.160) → κ →∞ 1100 17 Scalar Quantum Electrodynamics

More explicitly, the limit is

r/κ h(r) α πκ/2re− ., r . (17.161) → →∞ q 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 our natural units. Therefore, the behavior (17.160) holds very close to the origin. We can determine α by matching (17.160) to (17.156) from which we ﬁnd (using the small-r behavior K′ (r)= K (r) l/r): 0 − 1 ≈ − n α = . (17.162) κ In general, h(r) and ρ(r) have to be evaluated numerically. A typical solution for n = 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 equations of motion (17.88) and (17.89), and subtracting the condensation energy, we obtain from (17.131)

1 3 1 4 2 ∆F¯vort = F¯vort F¯c = d x (1 ρ )+ h . (17.163) − 2 Z 2 − 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 for ≈ r κ is given by (17.160). Inserting this into (17.151) with (17.158), we ﬁnd ≤ n2 ρ(r) 1 . (17.164) ≈ − 2r2 Thus the region 1 r κ yields an energy per unit length ≤ ≤ κ κ 1 ¯ 1 1 4 2 2 1 1 2 r Fvort = 2π drr (1 ρ )+ h = πn drr 2 + 2 K0 . (17.165) L 2 Z1 2 − Z1 r κ κ 17.7 Single Vortex Line and Critical Field Hc1 1101

For κ , the second integral becomes a constant, as a consequence of the integral →∞2 0∞ dxxK0 (x)=1/2. The first integral, on the other hand, has a logarithmic Rdivergence, so that we find 1 F¯ πn2[log κ + const.]. (17.166) L vort ≈ 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 G¯vort/L ext the magnetic G¯vort/L coupling hh so that, per unit length,

1 2 ∝ ext G¯h = πn (log κ +0.08) 2π drrhh . (17.167) L − Z0 But the integral over h is simply the flux quantum (17.144) associated with the vortex line, such that 1 G¯ = πn2(log κ +0.08) 2πnκ hext. (17.168) L h − When this quantity is smaller than zero, a vortex line invades the superconductor along the z-axis. The associated critical magnetic field is, in natural units, n log κ +0.08 h = . (17.169) c1 2 κ For large κ, this field can be quite small. In Fig. 17.3 we compare the asymptotic result with a numerical solution of the differential equation for n = 1, 2, 3,... in Fig. 17.3.

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. 1102 17 Scalar Quantum Electrodynamics

For a comparison with experiment one expresses this ﬁeld in terms of the critical magnetic ﬁeld hc =1/√2 and measures the ratio H n c1 = (log κ +0.08). (17.170) Hc √2κ

If the magnetic field is increased above Hc1, more and more flux tubes invade the type-II superconductor. These turn out to repel each other. The repulsive energy energy between them can be minimized if the flux tubes form a hexagonal array, as shown in Fig. 17.4. The tubes will perform thermal fluctuations, which may be so violent that the superflow experiences dissipation. The taming of these fluctuations is one of the main problems in high-temperature superconductors. It is usually done by 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 superconductor. Experimentally this is not quite true. The symmetric field inside the sample can 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 more general 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 then behaves 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 of the vector potential a which differ by gauge transformations. A convenient form is h a = (0, x, 0) . (17.173) κ Then ψ satisfies the Schrödinger equation

2 2 h 2 ∂x + i∂y + x ∂z ψ = ψ. (17.174) − κ ! −    This may be solved by a general ansatz

ψ(x, y, z)= eikzz+ikyyχ(x), (17.175) where χ(x) satisﬁes the diﬀerential equation

2 2 d h 2 + ky + x 1 kz χ(x)=0. (17.176) −dx2  − κ ! − −      This is the Schr¨odinger equation of a linear oscillator of frequency ω = h/κ, with the potential centered at κk x = y . (17.177) 0 h In fact, Eq. (17.176) can be written as

χ (x)+ ω2 (x x )2 χ(x)=(1 k 2)χ(x). (17.178) − xx − 0 − z This equation has a solution χ(x), which goes to zero for x , if →∞ 1 1 h 1 1 k2 E = ω n + = n + . (17.179) 2 − z ≡ n 2 κ 2 1 2 The energy En = 2 (1 kz ) is bounded by 1 from above. Hence, there cannot be solutions for arbitrarily− large h. The highest h is supported by the ground state solution, where n = 0. But also this ceases to exist as h reaches the critical value hc2 given by ω h 1 = = , (17.180) 2 2κ 2 where

hc2 = κ. (17.181)

Going back to physical magnetic ﬁelds, this amounts to

Hc2 = κ√2Hc. (17.182) 1104 17 Scalar Quantum Electrodynamics

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. q 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

1 ↔ 2 2 h j = ψ∗ ψ ψ a = χ(x) (0,ky,kz) (0, x, 0) 2i ∇ −| | | | " − κ #

2 h = χ(x) 0, (x x0),kz . (17.183) −| | κ − !

The superconducting sheet is centered around the plane with x = x0 parallel to the y z-plane. It carries no net charge flow along the x-axis. It consists of a double layer of 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 reduced to a lower value h κ 1 k2 . (17.184) c2 → − z The sheet configuration is obviously degenerate with respect to the position x0 of the sheet. Moreover, being in the regime of a linear Schrödinger equation, we can use the superposition principle to set up different spatial structures. For example, we may form an average over sheets in the rotated (cos ϕxˆ + sin ϕy,ˆ zˆ)-planes, and obtain cylindrical structures. These have a definite angular momentum around the z-axis rather than a definite linear momentum ky. Their wave function can be found directly from a vector potential h a = ( y, x, 0), (17.185) 2κ − which reads in cylindrical coordinates h a = ρ, a =0, a =0. (17.186) ϕ 2κ r z The linearized field equation now becomes 1 ∂ ∂ ∂2 1 ∂2 h2 r + + + r2 = ψ(z,r,ϕ). (17.187) "− r ∂r ∂r ∂z2 r2 ∂ϕ2 ! κ2 # It may be solved by a factorized ansatz

ψ(zrϕ)= χ( h/κr)eimϕeikzz. (17.188) q The diﬀerential operator in (17.187) is the same as for a two-dimensional harmonic oscillator of frequency ω = h/κ, and the solutions are well known. They may be expressed in terms of the conﬂuent hypergeometric functions 1F1(a; b; z) as

r′/2 m /2 χ(r′)= e− r′| | F ( n; m + 1; r′), (17.189) 1 1 − | | 17.8 Critical Field Hc2 where Superconductivity is Destroyed 1105

where r′ = h/κr. The corresponding energy eigenvalues are q 1 1 1 2 E = ω n + m m + + kz . (17.190) 2| | − 2 2 2 They have to be equated with 1/2(1 kz ) to obtain the highest possible magnetic ﬁeld, which is again found to be −

h = κ(1 k2), (17.191) c2 − z 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 y x h j = χ2 h/κr m , , 0 + (0, 0,k ) ( y, x, 0) r2 −r2 z − 2κ − q " # m h = χ2 h/κr k e + e . (17.192) z z r2 − 2κ ϕ q " ! # The wave function is non-zero within a narrow tube of diameter κ/h in units of the coherence length. Within this tube, a current is rotating clocqkwise around the z-axis, just as deduced before in the qualitative discussion of Eq. (17.150). In addition, 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 smaller than zero because of the negative interfacial energy. Thus, for H below Hc2, there will be states with nonzero magnetic field. If the magnetic field is far below Hc2, the magnitude ρ(x) of the order field will be so large that the quartic terms in the field equation (17.88) can no longer be neglected. The field configuration becomes inaccessible to simple analytic calcula- tions. The different critical magnetic fields for various superconducting materials are listed in Table 17.1.

Table 17.1 Different critical magnetic fields, in units of gauss, for various superconducting materials with different impurities.

material Hc Hc1 Hc2 Tc/K Pb 550 550 550 4.2 0.85 Pb, 0.15 Ir 650 250 3040 4.2 0.75 Pb, 0.25 In 570 200 3500 4.2 0.70 Pb, 0.30 Tl 430 145 2920 4.2 0.976 Pb, 0.024 Hg 580 340 1460 4.2 0.912 Pb , 0.088 Bi 675 245 3250 4.2 Nb 1608 1300 2680 4.2 0.5 Nb, 0.5 Ta 252 — 1370 5.6 1106 17 Scalar Quantum Electrodynamics

17.9 Order of Superconductive Phase Transition

The Ginzburg-Landau parameter κ does not only distinguish type I from type II superconductors. Its magnitude distinguishes also the order of the superconductive phase transition. If κ is very large, the effect of the electromagnetic field is extremely weak and the superconductor behaves very similar to a pure superfluid. Then the transition is certainly of second order. In the opposite limit of small κ, the mass acquired by the electromagnetic field is quite large and the penetration depth is very small. Then the transition becomes of first order. Somewhere between these limits, the order must change and it is possible to show that this happens at a tricritical 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 elementary excitations are the vortex lines whose grand-canonical ensemble is described by a fluctuating field. When vortex lines proliferate, the disorder field acquires a nonzero expectation 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 phase transition, marking a tricritical point.

17.10 Quartic Interaction and Tricritical Point

According to the action (17.9), the Ginzburg-Landau theory of superconductivity is characterized by the following free energy density: 1 f(ϕ, ϕ, A, A) = ( + iqA) ϕ∗ ( iqA) ϕ ∇ ∇ 2 ∇ ∇ − m2 g + ϕ 2 + ϕ 4 (17.193) 2 | | 4| | 1 + ( A)2 , 2 ∇ × with the order field iγ(x) ϕ(x)= ρ(x)e− , (17.194) where ρ(x) and γ(x) are real variables. The electromagnetic field is represented by the vector A, and q = 2e/hc¯ accounts for the electromagnetic coupling of Cooper pairs whose charge is 2e.3 The real constants m2 and g parametrize the size of the quadratic and quartic terms, respectively. If the mass parameter m2 drops below zero, the ground state of the potential 1 g V (ϕ)= m2 ϕ 2 + ϕ 4 (17.195) 2 | | 4| | is obtained by an infinite number of degenerate states that satisfy m2 ϕ 2 = ρ2 = . (17.196) h i 0 − g 3The Euler number is represented by the roman letter e, not to be confused with the electric charge 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

1 q2ρ2 f(ϕ, ϕ, A, A) = ( ρ)2 + V (ρ)+ A2 ∇ ∇ 2 ∇ 2 1 + ( A)2 . (17.197) 2 ∇ × 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 coherence length ξ =1/√ 2m2 and the penetration depth λ , constitutes a dimensionless − L material parameter of a superconductor, the so-called Ginzburg-Landau parameter κ λ /√2ξ. Using (17.196) we ﬁnd for κ the value ≡ L g κ = . (17.199) sq2

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 2 2 cubic term. For m = mt , there exist two minima of equal height V (ρ), one at ρ = 0 and 2 2 2 another one around ρt, causing a first-order transition at m = mt . When m 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 2 mf . 1108 17 Scalar Quantum Electrodynamics

1 g c q3 V (ρ)= m2ρ2 + ρ4 ρ3 , c = . (17.200) 2 4 − 3 2π We can see in Fig. 17.5 that the cubic term generates for m2 < c2/4g a second minimum at c 4m2g ρ = 1+ 1 . (17.201) 2 2g  s − c2 

2 2   At the specific point mt =2c /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 of first-order (tricritical point). Therefore, in this point, the coherence length of the ρ-field fluctuations becomes 1 3 g ξt = = . (17.202) m2 +3gρ2 2cρ c r 2 t − t q 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 goes over into type II superconductivity at [4, 5, 6]

3√3 4 π 0.8 κ κt = 1 . (17.203) ≈ 2π s − 9 3 ≈ √2 If the mass term drops to zero, the second minimum, at which the ﬂuctuations stabilize, has a curvature that determines a square mass

2 mf =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- iγ(x) Landau theory, the Coleman-Weinberg model. Setting again ϕ(x) = ρ(x)e− as in (17.194) (see also the O(N)-model in Chapter 22 for N = 2), the eﬀective potential of the Coleman-Weinberg model at one-loop level is [9]

1 g 3q4 ρ2 25 V (ρ)= m2ρ2 + ρ4 + ρ4 log , (17.205) 2 4 64π2 µ2 − 6 ! 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 the renormalization is done. We shall assume g to be of the same order as q4, which is very 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 by setting ∝ g 3q4 µ2 11 = log + , (17.206) 4 64π2 M 2 3 ! which turns the effective potential into

1 3q4 ρ2 1 V (ρ)= m2ρ2 + ρ4 log . (17.207) 2 64π2 M 2 − 2!

If the theory is massless, i.e., if m2 = 0, the potential (17.207) has a minimum at the ﬁeld value ρ = ρ M 2. (17.208) f ≡ 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 2 2 ρt causing a first-order transition for m = mt . For m = 0, the effective potential has a 2 minimum at some ρ = ρf > ρt with a fluctuation-generated mass term mf .

The curvature implies a nonzero mass generated by ﬂuctuations:

2 4 2 2 4 2 2 2 ∂ V q ρf ρf 6q M mf = m (ρf )= = 9 log +6 = . (17.209) ∂ρ2 16π2 M 2 ! 16π2 m=0,ρ=ρc

As described in [9, 10], the eﬀective potential has, for a positive m2, both a maximum 2 2 1/2 4 2 1 2 and a minimum as long as m < 2m1/e 3q M e− /16π . The minimum of the ≡ 2 2 4 2 1/2 2 potential lies at the same level as the origin if m = mt 3q M e− /32π with 2 2 1/2 ≡ ρt = M e− . This is the tricritical point, and the scalar mass at that point is

2 4 2 2 4 2 2 ∂ V 2 q ρt ρt 3q M m (ρt) = = mt + 9 log +6 = ∂ρ2 16π2 M 2 ! 16π2e1/2 ρ=ρt

λ = ρ2, (17.210) α t 1110 17 Scalar Quantum Electrodynamics where µ2 11 α = log + (17.211) M 2 3 ! 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 two characteristic length scales of the theory [compare (17.199)]

λL 1 m(ρt) 1 g κ = = = 2 . (17.212) √2ξ √2 qρt √2α sq

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 the renormalization scale M. The above U(1)-gauge-invariant framework is still far from the full description of the electroweak interaction, where the true symmetry group is SU(2)L U(1)Y (see Chapter 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 illustrates how fluctuations are capable of spontaneously converting a classically massless theory into a massive theory. This is best seen by looking at the set of effective classical potentials for various mass terms m2ρ2/2, shown in Fig. 17.6. If m2 = 0, the curvature of the quadratic term in the field ∂V (ρ)/∂ρ2 at ρ = 0 is zero and the theory is massless. The effect is that the field fluctuations around ρ = 0 diverge and make the theory critical. The interactions of the field limit the fluctuations so that the field expectation settles at some value ρf = M > ρt with a finite mass mf . From Eq.(17.210) we see that

3q4M 2 m2 = . (17.213) f 8π2 Remember that after the ρ-field acquires a nonzero expectation, the vector field has a nonzero mass mA = qM given by Eq. (17.198). Inserting this into (17.213), we obtain the famous experimentally observable mass ratio caused by fluctuations of a massless theory:

2 2 mf 3 q 2 = . (17.214) mA 2π 4π The value of the eﬀective potential at this new minimum is, according to Eq. (17.207),

3q4M 4 3m4 V (ρ )= = A . (17.215) f − 128π2 −128π2 Notes and References 1111

This is the important lesson of the Coleman-Weinberg model. Even though the classical theory has a scalar field of zero mass, the fluctuations produce a nonzero mass mf via the nonzero field expectation value ρf = M. The mass mf is a so-called spontaneously generated mass. The origin of the mass lies in the need to introduce some nonzero mass scale µ when calculating divergent loop diagrams. The new finite mass parameter M 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 textbooks D. 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).