Carsten Timm: Theory of superconductivity

∇ϕ − (q/c)A. Physics is invariant under the gauge transformation
A → A + ∇χ, (6.59)
Φ → Φ − 1 c
˙χ, (6.60)
ψ → e iqχ/c ψ, (6.61)
where Φ is the scalar electric potential and χ(r, t) is an arbitrary scalar field. We make use of this gauge invariance
by choosing
χ = − c ϕ. (6.62)
q
Under this tranformation, we get
A → A − c
q ∇ϕ =: A′ , (6.63)
ψ = (ψ 0 + δψ) e iϕ → (ψ 0 + δψ) e i(−ϕ+ϕ) = ψ 0 + δψ. (6.64)
The macroscopic wave function becomes purely real (and positive). The Landau functional thus tranforms into
∫ [
F [δψ, A ′ ] → d 3 r − 2αδψ 2 + 1 ( ) ∗
2m ∗ i ∇δψ ·
i ∇δψ
− α q 2
β 2m ∗ c 2 (A′ ) ∗ · A ′ + 1
]
8π (∇ × A′ ) ∗ · (∇ × A ′ )
(6.65)
(note that ∇×A ′ = ∇×A). Thus the phase no longer appears in F ; it has been absorbed into the vector potential.
Furthermore, dropping the prime,
F [δψ, A] ∼ = ∑ [(−2α + 2 k 2 )
2m ∗ δψkδψ ∗ k − α q 2
β 2m ∗ c 2 A∗ k · A k + 1
]
8π (k × A k) ∗ · (k × A k )
k
= ∑ [(−2α + 2 k 2 )
2m ∗ δψkδψ ∗ k − α q 2
β 2m ∗ c 2 A∗ k · A k + k2
8π A∗ k · A k − 1
]
8π (k · A∗ k)(k · A k ) . (6.66)
k
Obviously, amplitude fluctuations decouple from electromagnetic fluctuations and behave like for a neutral superfluid.
We discuss the electromagnetic fluctuations further. The term proportional to α is due to superconductivity.
Without it, we would have the free-field functional
F free [A] = ∑ k
1 [
k 2 A ∗ k · A k − (k · A ∗
8π
k)(k · A k ) ] . (6.67)
Decomposing A into longitudinal and transverse components,
A k = ˆk(ˆk · A k ) + A k −
} {{ }
ˆk(ˆk · A k )
} {{ }
=: A ⊥ k
=: A ∥ k
(6.68)
with ˆk := k/k, we obtain
F free [A] = ∑ k
= ∑ k
1
[
(
k 2 A ∥∗
k
8π · A∥ k + k2 A ⊥∗
k · A ⊥ k −
[✘✘✘✘ ✘
1
8π
k 2 A ∥∗
k
· A∥ k + k2 A ⊥∗
k
k · A ∥∗
k
) ( )]
k · A ∥ k
· A ⊥ k − ✘ k 2 ✘✘✘
A ∥∗ ✘ ]
k · A∥ k
= ∑ k
1
8π k2 A ⊥∗
k · A ⊥ k . (6.69)
Thus only the transverse components appear—only they are degrees of freedom of the free electromagnetic field.
They do not have an energy gap.
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