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1 1
(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)


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

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