Carsten Timm: Theory of superconductivity

and
−V RPA
eff ≡ := + + + · · · (8.80)
or
As above, we can sum this up,
−V RPA
eff (q, iν n ) = −V eff (q, iν n ) + V eff (q, iν n ) Π 0 (q, iν n ) V eff (q, iν n )
− V eff (q, iν n ) Π 0 (q, iν n ) V eff (q, iν n ) Π 0 (q, iν n ) V eff (q, iν n ) + · · · (8.81)
Veff RPA
V eff (q, iν n )
(q, iν n ) =
1 + V eff (q, iν n )Π 0 (q, iν n ) = V C(q)
= V C (q)
=
1 + V C (q)
(iν n ) 2
(iν n ) 2 − Ω 2 + (iν n ) 2 V C (q)Π 0 (q, iν n )
V C (q)
1 + V C (q)Π 0 (q, iν n )
} {{ }
= VC RPA (q,iν n)
= V RPA
C (q, iν n )
(iν n ) 2
(iν n ) 2 −Ω 2
(iν n ) 2
(iν n ) 2 −Ω 2 Π 0 (q, iν n )
(iν n ) 2 + (iν n ) 2 V C (q)Π 0 (q, iν n )
(iν n ) 2 − Ω 2 + (iν n ) 2 V C (q)Π 0 (q, iν n )
(iν n ) 2
(iν n ) 2 −
Ω 2
1+V C (q)Π 0 (q,iν n )
= VC RPA
(iν n ) 2
(q, iν n )
(iν n ) 2 − ωq(iν 2 n )
(8.82)
with the renormalized phonon frequency
ω q (iν n ) :=
Ω
√
1 + VC (q)Π 0 (q, iν n ) . (8.83)
To see that this is a reasonable terminology, compare V RPA
eff
V eff (q, iν n ) = V C (q)
to the bare effective interaction
(iν n ) 2
(iν n ) 2 − Ω 2 . (8.84)
Evidently, screening leads to the replacements V C → VC RPA and Ω → ω q .
For small momenta and frequencies, we have Π 0 → N(E F ), the density of states at E F . In this limit we thus
obtain
ω q
∼ =
Ω
√
1 + 4π e2
q
N(E 2 F )
=
Ω
√
1 + κ2 s
√
κ 2 s
q 2 ∼ =
Ω
q 2 = Ω κ s
q. (8.85)
Due to screening we thus find an acoustic dispersion of jellium phonons. This is of course much more realistic
than an optical Einstein mode.
Beyond the low-frequency limit it is important that Π 0 and thus ω q obtains a sizable imaginary part. It smears
out the pole in the retarded interaction Veff
RPA (q, ν) or rather moves it away from the real-frequency axis—the
lattice vibrations are now damped. The real part of the retarded interaction is sketched here for fixed q:
80