Carsten Timm: Theory of superconductivity

(the minus sign is conventional) and
−V full
C ≡ (8.45)
as well as
G 0 ≡ (8.46)
for the bare electronic Green function one would find for the non-interacting Hamiltonian H 0 , we obtain
=
+ + +
+ + + +
+ · · · (8.47)
This is an expansion in powers of e 2 since V C contributes a factor of e 2 . We have exhibited all diagrams up to
order e 6 . If we try to evaluate this sum term by term, we accounter a problem: At each vertex •, momentum and
energy (frequency) must be conserved. Thus in partial diagrams of the form
k
k
q = 0
the Coulomb interaction carries momentum q = 0. But
V C (0) = 4π e2
0 2 (8.48)
is infinite. However, one can show that the closed G loop corresponds to the average electron density so that the
diagram signifies the Coulomb interaction with the average electronic charge density (i.e., the Hartree energy).
But this is compensated by the average charge density of the nuclei. Thus we can omit all diagrams containing
the “tadpole” diagram shown above.
Since we still cannot evaluate the sum in closed form we need an approximation. We first consider the limiting
cases of small and large q in VC
full(q,
iν n).
• For large q, corresponding to small distances, the first diagram is proportional to 1/q 2 , whereas all the
others are at least of order 1/q 4 and are thus suppressed. We should recover the bare Coulomb interaction
for large q or small distances, which is plausible since the polarization of the electron gas cannot efficiently
screen the interaction between two test charges that are close together.
• For small q we find that higher-order terms contain higher and higher powers of 1/q 2 and thus become
very large. This is alarming. The central idea of our approximation is to keep only the dominant term
(diagram) at each order in e 2 . The dominant term is the one with the highest power in 1/q 2 . Only the
V C lines forming the backbone of the diagrams (drawn horizontally) carry the external momentum q due
to momentum conservation at the vertices. Thus the dominant terms are the ones with all V C lines in the
backbone:
q
k
q
k’
q
k+ q
k’ + q
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