Carsten Timm: Theory of superconductivity

Summing up these dominant terms we obtain the approximate effective interaction
−V RPA
C := +
+ +
+ · · · (8.49)
This is called the random phase approximation (RPA) for historical reasons that do not concern us here, or the
Thomas-Fermi approximation. The most important part of RPA diagrams is clearly the bubble diagram
Π 0 ≡ , (8.50)
which stands for
Π 0 (q, iν n ) = − 1 β
= − 2 β
∑ 1
V
iω n
∑ ∑
iω n
k
∑
Gk+q,σ(iω 0 n + iν n ) Gkσ(iω 0 n )
kσ
1
iω n + iν n − ξ k+q
1
iω n − ξ k
(8.51)
(the factor of 2 is due to the spin). We can write the diagramatic series also as
−V RPA
C (q, iν n ) = −V C (q) + V C (q) Π 0 (q, iν n ) V C (q) − V C (q) Π 0 (q, iν n ) V C (q) Π 0 (q, iν n ) V C (q) + · · ·
= −V C (q) [ 1 − Π 0 (q, iν n ) V C (q) + Π 0 (q, iν n ) V C (q) Π 0 (q, iν n ) V C (q) − · · · ].
(8.52)
This is a geometric series, which we can sum up,
VC RPA
V C (q)
(q, iν n ) =
1 + V C (q) Π 0 (q, iν n ) . (8.53)
Π 0 can be evaluated, it is essentially −1 times the susceptibility of the free electron gas. We cannot explain this
here, but it is plausible that the susceptibility, which controls the electric polarization of the electron gas, should
enter into a calculation of the screened Coulomb interaction. Note that VC
RPA has a frequency dependence since
Π 0 (or the susceptibility) has one.
In the static limit iν → 0 + i0 + and at low temperatures T ≪ E F /k B , one can show that
Π 0 (q, 0) ∼ = const = N(E F ), (8.54)
where N(E F ) is the electronic density of states at the Fermi energy, including a factor of two for the spin. Thus
we obtain
V RPA
C (q) =
4π e2
q 2
e 2
= 4π
1 + 4π e2
q
Π 2 0 q 2 + 4πe 2 Π 0
e 2
∼= 4π
q 2 + 4πe 2 N(E F )
e 2
= 4π
q 2 + κ 2 s
(8.55)
with
κ s := √ 4πe 2 N(E F ). (8.56)
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