Carsten Timm: Theory of superconductivity

Including the superconducting contribution, we get
F [A] = ∑ [
− α q 2
β 2m ∗ c 2 A∥∗ k · A∥ k − α q 2
β 2m ∗ c 2 A⊥∗ k · A ⊥ k + 1
]
8π k2 A ⊥∗
k · A ⊥ k . (6.70)
k
All three components of A appear now, the longitudinal one has been introduced by absorbing the phase ϕ(r).
Even more importantly, all components obtain a term with a constant coefficient, i.e., a mass term. Thus the
electromagnetic field inside a superconductor becomes massive. This is the famous Anderson-Higgs mechanism.
The same general idea is also thought to explain the masses of elementary particles, although in a more complicated
way. The “Higgs bosons” in our case are the amplitude-fluctuation modes described by δψ. Contrary to what is
said in popular discussions, they are not responsible for giving mass to the field A. Rather, they are left over
when the phase fluctuations are eaten by the field A.
The mass term in the superconducting case can be thought of as leading to the Meißner effect (finite penetration
depth λ). Indeed, we can write
F [δψ, A] = ∑ k
(−2α + 2 k 2
2m ∗ )
δψ ∗ kδψ k + ∑ k
[
1 1
8π λ 2 A∗ k · A k + k 2 A ∗ k · A k − (k · A ∗ k)(k · A k )]
. (6.71)
The photon mass is proportional to 1/λ. (To see that the dispersion relation is 2 ω 2 = m 2 c 4 + p 2 c 2 , we would
have to consider the full action including a temporal integral over F and terms containing time-derivatives.)
Elitzur’s theorem
One sometimes reads that in the superconducting state gauge symmetry is broken. This is not correct. Gauge
symmetry is the invariance under local gauge transformations. S. Elitzur showed in 1975 that a local gauge
symmetry cannot be spontaneously broken. Rather, superconductors and superfluids spontaneously break a
global U(1) symmetry in that the ordered state has a prefered macroscopic phase, as noted above.
6.4 Type-I superconductors
Superconductors with small Ginzburg-Landau parameter κ are said to be of type I. The exact condition is
κ < 1 √
2
. (6.72)
It turns out that these superconductors have a uniform state in an applied magnetic field, at least for simple
geometries such as a thin cylinder parallel to the applied field.
The appropriate thermodyamic potential for describing a superconductor in an applied magnetic field is the
Gibbs free energy G (natural variable H) and not the Helmholtz free energy F (natural variable B), which we
have used so far. They are related by the Legendre transformation
∫
G = F −
d 3 r H · B
4π . (6.73)
For a type-I superconductor, the equilibrium state in the bulk is uniform (we will see later why this is not a
trivial statement). The order parameter is then |ψ| = √ −α/β and the magnetic induction B as well as the vector
potential (in the London gauge) vanish. Thus the Gibbs free-energy density is
g s = f s = α |ψ| 2 + β 2 |ψ|4 = − α2
β + β α 2
2 β = − α2
2β . (6.74)
On the other hand, in the normal state ψ vanishes, but the field penetrates the system and B ≡ H so that
g n = f n − HB
4π = H2
8π − H2
4π = −H2 8π . (6.75)
41