(x) = (∇ρ)2 + B 2 + e2 ρ2 A2 + »(ρ2 ’ a2 )2 . (15)

2 4

In this unitary gauge, the residual gauge freedom in the vector potential has

disappeared together with the phase of the matter ¬eld. In addition to condi-

tion (11), ¬elds of vanishing energy must satisfy

A = 0, ρ = a. (16)

In small oscillations of the gauge ¬eld around the ground state con¬gurations (16)

a restoring force appears as a consequence of the non-vanishing value a of the

Higgs ¬eld ρ. Comparison with the energy density of a massive non-interacting

scalar ¬eld •

1 1 22

2

• (x) = (∇•) + M •

2 2

shows that the term quadratic in the gauge ¬eld A in (15) has to be interpreted

as a mass term of the vector ¬eld A. In this Higgs mechanism, the photon has

acquired the mass √

Mγ = 2ea , (17)

which is determined by the value of the Higgs ¬eld. For non-vanishing Higgs ¬eld,

the zero energy con¬guration and the associated small amplitude oscillations

describe electrodynamics in the so called Higgs phase, which di¬ers signi¬cantly

from the familiar Coulomb phase of electrodynamics. In particular, with photons

becoming massive, the system does not exhibit long range forces. This is most

directly illustrated by application of the abelian Higgs model to the phenomenon

of superconductivity.

Meissner E¬ect. In this application to condensed matter physics, one identi¬es

the energy density (15) with the free-energy density of a superconductor. This

is called the Ginzburg“Landau model. In this model |φ|2 is identi¬ed with the

density of the superconducting Cooper pairs (also the electric charge should be

14 F. Lenz

replaced e ’ e = 2e) and serves as the order parameter to distinguish normal

a = 0 and superconducting a = 0 phases.

Static solutions (11) satisfy the Hamilton equation (cf. (10), (15))

δH

= 0,

δA(x)

which for a spatially constant scalar ¬eld becomes the Maxwell“London equation

rot B = rot rot A = j = 2e2 a2 A .

The solution to this equation for a magnetic ¬eld in the normal conducting phase

(a = 0 for x < 0)

B(x) = B0 e’x/»L (18)

decays when penetrating into the superconducting region (a = 0 for x > 0)

within the penetration or London depth

1

»L = (19)

Mγ

determined by the photon mass. The expulsion of the magnetic ¬eld from the

superconducting region is called Meissner e¬ect.

Application of the gauge transformation ((7), (14)) has been essential for

displaying the physics content of the abelian Higgs model. Its de¬nition requires

a well de¬ned phase θ(x) of the matter ¬eld which in turn requires φ(x) = 0.

At points where the matter ¬eld vanishes, the transformed gauge ¬elds A

are singular. When approaching the Coulomb phase (a ’ 0), the Higgs ¬eld

oscillates around φ = 0. In the unitary gauge, the transition from the Higgs to

the Coulomb phase is therefore expected to be accompanied by the appearance

of singular ¬eld con¬gurations or equivalently by a “condensation” of singular

points.

2.2 Topological Excitations

In the abelian Higgs model, the manifold of ¬eld con¬gurations is a circle S 1

parameterized by the angle β in (13). The non-trivial topology of the manifold

of vacuum ¬eld con¬gurations is the origin of the topological excitations in the

abelian Higgs model as well as in the other ¬eld theoretic models to be discussed

later. We proceed as in the discussion of the ground state con¬gurations and

consider static ¬elds (11) but allow for energy densities which do not vanish

everywhere. As follows immediately from the expression (10) for the energy

density, ¬nite energy can result only if asymptotically (|x| ’ ∞)

’ aeiθ(x)

φ(x)

’0

B(x)

Dφ(x) = (∇ ’ ieA(x)) φ(x) ’ 0. (20)

Topological Concepts in Gauge Theories 15

For these requirements to be satis¬ed, scalar and gauge ¬elds have to be corre-

lated asymptotically. According to the last equation, the gauge ¬eld is asymp-

totically given by the phase of the scalar ¬eld

1 1

A(x) = ∇ ln φ(x) = ∇θ(x) . (21)

ie e

The vector potential is by construction asymptotically a “pure gauge” (8) and

no magnetic ¬eld strength is associated with A(x).

Quantization of Magnetic Flux. The structure (21) of the asymptotic gauge

¬eld implies that the magnetic ¬‚ux of ¬eld con¬gurations with ¬nite energy is

quantized. Applying Stokes™ theorem to a surface Σ which is bounded by an

asymptotic curve C yields

1 2π

A · ds = ∇θ(x) · ds = n

¦n = B d2 x = . (22)

B

e e

C C

Σ

Being an integer multiple of the fundamental unit of magnetic ¬‚ux, ¦n cannot

B

change as a function of time, it is a conserved quantity. The appearance of

this conserved quantity does not have its origin in an underlying symmetry,

rather it is of topological origin. ¦n is also considered as a topological invariant

B

since it cannot be changed in a continuous deformation of the asymptotic curve

C. In order to illustrate the topological meaning of this result, we assume the

asymptotic curve C to be a circle. On this circle, |φ| = a (cf. (13)). Thus the

scalar ¬eld φ(x) provides a mapping of the asymptotic circle C to the circle of

zeroes of the Higgs potential (V (a) = 0). To study this mapping in detail, it is

convenient to introduce polar coordinates

φ(x) = φ(r, •) r’∞ aeiθ(•) eiθ(•+2π) = eiθ(•) .

,

’’

The phase of the scalar ¬eld de¬nes a non-trivial mapping of the asymptotic

circle

θ : S 1 ’ S 1 , θ(• + 2π) = θ(•) + 2nπ (23)

to the circle |φ| = a in the complex plane. These mappings are naturally divided

into (equivalence) classes which are characterized by their winding number n.

This winding number counts how often the phase θ winds around the circle when

the asymptotic circle (•) is traversed once. A formal de¬nition of the winding

number is obtained by decomposing a continuous but otherwise arbitrary θ(•)

into a strictly periodic and a linear function

n = 0, ±1, . . .

θn (•) = θperiod (•) + n•

where

θperiod (• + 2π) = θperiod (•).

The linear functions can serve as representatives of the equivalence classes. El-

ements of an equivalence class can be obtained from each other by continuous

16 F. Lenz

Fig. 3. Phase of a matter ¬eld with winding number n = 1 (left) and n = ’1 (right)

deformations. The magnetic ¬‚ux is according to (22) given by the phase of the

Higgs ¬eld and is therefore quantized by the winding number n of the map-

ping (23). For instance, for ¬eld con¬gurations carrying one unit of magnetic

¬‚ux, the phase of the Higgs ¬eld belongs to the equivalence class θ1 . Figure 3

illustrates the complete turn in the phase when moving around the asymptotic

circle. For n = 1, the phase θ(x) follows, up to continuous deformations, the po-

lar angle •, i.e. θ(•) = •. Note that by continuous deformations the radial vector

¬eld can be turned into the velocity ¬eld of a vortex θ(•) = • + π/4. Because

of their shape, the n = ’1 singularities, θ(•) = π ’ •, are sometimes referred

to as “hyperbolic” (right-hand side of Fig. 3). Field con¬gurations A(x), φ(x)

with n = 0 are called vortices and possess indeed properties familiar from hy-

drodynamics. The energy density of vortices cannot be zero everywhere with the

magnetic ¬‚ux ¦n = 0. Therefore in a ¬nite region of space B = 0. Furthermore,

B

the scalar ¬eld must at least have one zero, otherwise a singularity arises when

contracting the asymptotic circle to a point. Around a zero of |φ|, the Higgs ¬eld

displays a rapidly varying phase θ(x) similar to the rapid change in direction

of the velocity ¬eld close to the center of a vortex in a ¬‚uid. However, with the

modulus of the Higgs ¬eld approaching zero, no in¬nite energy density is asso-

ciated with this in¬nite variation in the phase. In the Ginzburg“Landau theory,

the core of the vortex contains no Cooper pairs (φ = 0), the system is locally in

the ordinary conducting phase containing a magnetic ¬eld.

The Structure of Vortices. The structure of the vortices can be studied in

detail by solving the Euler“Lagrange equations of the abelian Higgs model (2).

To this end, it is convenient to change to dimensionless variables (note that in

2+1 dimensions φ, Aµ , and e are of dimension length’1/2 )

1 1 1 »

x’ A’ φ’

x, A, φ, β= . (24)

2e2

ea a a

Accordingly, the energy of the static solutions becomes

E 1 β

d2 x |(∇ ’ iA)φ| + (∇ — A)2 + (φφ— ’ 1)2

2

= . (25)

a2 2 2

The static spherically symmetric Ansatz

±(r)

φ = |φ(r)|ein• , A=n e• ,

r

Topological Concepts in Gauge Theories 17

converts the equations of motion into a system of (ordinary) di¬erential equations

coupling gauge and Higgs ¬elds

d2 n2

1d 2

’ 2’ |φ| + 2 (1 ’ ±) |φ| + β(|φ|2 ’ 1)|φ| = 0 , (26)

dr r dr r

d2 ± 1 d±

’ ’ 2(± ’ 1)|φ|2 = 0 . (27)

2

dr r dr

The requirement of ¬nite energy asymptotically and in the core of the vortex

leads to the following boundary conditions

r ’ ∞ : ± ’ 1 , |φ| ’ 1 , ±(0) = |φ(0)| = 0. (28)

From the boundary conditions and the di¬erential equations, the behavior of

Higgs and gauge ¬elds is obtained in the core of the vortex

± ∼ ’2r2 , |φ| ∼ rn ,

and asymptotically

√ √

√ √

’ 2r ’ 2βr

±’1∼ |φ| ’ 1 ∼

re , re .

The transition from the core of the vortex to the asymptotics occurs on di¬erent

scales for gauge and Higgs ¬elds. The scale of the variations in the gauge ¬eld

is the penetration depth »L determined by the photon mass (cf. (18) and (19)).

It controls the exponential decay of the magnetic ¬eld when reaching into the

superconducting phase. The coherence length

1 1

√ =√

ξ= (29)

ea 2β a»

controls the size of the region of the “false” Higgs vacuum (φ = 0). In super-

conductivity, ξ sets the scale for the change in the density of Cooper pairs. The

Ginzburg“Landau parameter

»L

κ= = β (30)

ξ

varies with the substance and distinguishes Type I (κ < 1) from Type II (κ > 1)

superconductors. When applying the abelian Higgs model to superconductivity,

one simply reinterprets the vortices in 2 dimensional space as 3 dimensional ob-

jects by assuming independence of the third coordinate. Often the experimental

setting singles out one of the 3 space dimensions. In such a 3 dimensional inter-

pretation, the requirement of ¬nite vortex energy is replaced by the requirement

of ¬nite energy/length, i.e. ¬nite tension. In Type II superconductors, if the

strength of an applied external magnetic ¬eld exceeds a certain critical value,

magnetic ¬‚ux is not completely excluded from the superconducting region. It

penetrates the superconducting region by exciting one or more vortices each of

18 F. Lenz

which carrying a single quantum of magnetic ¬‚ux ¦1 (22). In Type I supercon-

B

ductors, the large coherence length ξ prevents a su¬ciently fast rise of the Cooper

pair density. In turn the associated shielding currents are not su¬ciently strong

to contain the ¬‚ux within the penetration length »L and therefore no vortex can

form. Depending on the applied magnetic ¬eld and the temperature, the Type II

superconductors exhibit a variety of phenomena related to the intricate dynam-

ics of the vortex lines and display various phases such as vortex lattices, liquid

or amorphous phases (cf. [11,12]). The formation of magnetic ¬‚ux lines inside

Type II superconductors by excitation of vortices can be viewed as mechanism

for con¬ning magnetic monopoles. In a Gedankenexperiment we may imagine to

introduce a north and south magnetic monopole inside a type II superconductor

separated by a distance d. Since the magnetic ¬eld will be concentrated in the

core of the vortices and will not extend into the superconducting region, the ¬eld

energy of this system becomes

1 4πd

d 3 x B2 ∝

V= . (31)

e2 »2