6
0 0.5 10
0.5
1
ω  in eV
σ ( ω )
Conductivities (BCSI)
NS
SC
2 ∆
0 50 100160
180
200
Γ  in meV
W K  in meV
BCSI
SC
NS
FIG. 4: Top - a conductivity plot for the BCSI case in the
presence of a lattice. The parameters are ∆ = 30meV , Γ =
3.5 meV . Bottom – the behavior of Kubo sums. Note that (a)
the spectral weight in the NS is always greater in the SCS, (b)
the spectral weight decreases with Γ, and (c) the diﬀerence
between NS and SCS decreases as Γ increases.
little variation of ∆W (ωc) at above 0 .1 − 0.3eV what
implies that for larger ωc, ∆ W (ωc) ≈ ∆ WK >> ∆ f(ωc).
To make this more quantitative, we compare in Fig. 6
∆ W (ωc) obtained for a constant DOS, when ∆ W (ωc) =
∆ f(ωc), and for the actual lattice dispersion, when
∆ W (ωc) = ∆ WK + ∆ f(ωc). In the clean limit there
is obviously little cutoﬀ dependence beyond 0 .1eV , i.e.,
∆ f(ωc) is truly small, and the diﬀerence between the
two cases is just ∆ WK . In the dirty limit, the situation
is similar, but there is obviously more variation with ωc,
and ∆ f(ωc) becomes truly small only above 0 .3eV . Note
also that the position of the dip in ∆ W (ωc) in the clean
limit is at a larger ωc in the presence of the lattice than
in a continuum.
B. The Einstein boson model
We next consider the case of electrons interacting with
a single boson mode which by itself is not aﬀected by su-
perconductivity. The primary candidate for such mode is
an optical phonon. The imaginary part of the NS self en-
ergy has been discussed numerous times in the literature.
We make one simplifying assumption – approximate the
DOS by a constant in calculating fermionic self-energy.
We will, however, keep the full lattice dispersion in the
calculations of the optical integral. The advantage of this
0 0.5 10  
0.5
1  
ω c  in eV
W( ω c )/W( ∞ )
Normal State Optical Sum (BCSI)
Dirty Limit
Clean Limit
0 0.5 10  
0.5
1  
ω c  in eV
W( ω c )/W( ∞ )
Superconducting State Optical Sum (BCSI)
Dirty Limit
Clean Limit
FIG. 5: The evolution of optical integral in NS(top) and
SCS(bottom) for BCSI case. Plots are made for clean limit
(solid lines, Γ = 3.5 meV ) and dirty limit (dashed lines,
Γ = 150 meV ) for ∆ = 30 meV . Observe that (a) W (0) = 0
in the NS, but has a non-zero value in the SCS because of the
δ-function (this value decreases in the dirty limit), and (b)
the ﬂat region in the SCS is due to the fact that σ′(ω) = 0 for
Ω < 2∆. Also note that ∼ 90 − 95% of the spectral weight is
recovered up to 1 eV
approximation is that the self-energy can be computed
analytically. The full self-energy obtained with the lat-
tice dispersion is more involved and can only be obtained
numerically, but its structure is quite similar to the one
obtained with a constant DOS.
The self-energy for a constant DOS is given by
Σ( iω) = − i
2π λn
∫
dǫkd(iΩ) χ(iΩ) G(ǫk, iω + iΩ) (13)
where
χ(iΩ) = ω2
0
ω2
0 − (iΩ) 2 (14)
and λn is a dimensionless electron-boson coupling. Inte-
grating and transforming to real frequencies, we obtain
Σ ′′(ω) = − π
2 λnωo Θ( |ω| − ωo)
Σ ′(ω) = − 1
2 λnωo log
⏐
⏐
⏐
⏐
ω + ωo
ω − ωo
⏐
⏐
⏐
⏐ (15)
In the SCS, we obtain for ω < 0
Σ ′′(ω) = − π
2 λnωo Re
(
ω + ωo√
(ω + ωo)2 − ∆ 2
)