7
0 0.1 0.2 0.3
0 
20
40
ω c  in eV
W SC ( ω c ) − W NS ( ω c )
∆  W (BCSI−clean limit)
 with lattice
without lattice
0.1 0.3 0.5
0 
10
ω c  in eV
W SC ( ω c ) − W NS ( ω c )
∆  W (BCSI−dirty limit)
0.5 1
−1
0
ω c  in eV
∆  W
Larger ω c
FIG. 6: Evolution of ∆ W in the presence of a lattice (solid
line) compared with the case of no lattice(a constant DOS,
dashed line) for clean and dirty limits. ∆ = 30meV , Γ =
3.5 meV (clean limit), Γ = 150 meV (dirty limit)
Σ ′(ω) = − 1
2 λnωo Re
∫
dω′ 1
ω2o − ω′2 − iδ
ω + ω′
√
(ω + ω′)2 − ∆ 2
(16)
Observe that Σ ′′(ω) is no-zero only for ω < −ωo − ∆.
Also, although it does not straightforwardly follow from
Eq. 16, but real and imaginary parts of the self-energy
do satisfy Σ′(ω) = −Σ ′(−ω) and Σ ′′(ω) = Σ ′′(−ω).
Fig7 shows conductivities σ(ω) and Kubo sums WK
as a function of the dimensionless coupling λ. We see
that, like in the previous case, the Kubo sum in the NS
is larger than that in the SCS. The diﬀerence ∆WK is
between 5 and 8 meV.
Fig 8 shows the evolution of the optical integrals. Here
we see the diﬀerence with the BCSI model – only about
75% of the optical integral is recovered, both in the NS
and SCS, when we integrate up to the bandwidth of 1eV .
The rest comes from higher frequencies.
In Fig 9 we plot ∆ W (ωc) as a function of ωc. We see
the same behavior as in the BCSI model in a clean limit
– ∆W (ωc) is positive at small frequencies, crosses zero
at some ωc, passes through a deep minimum at a larger
frequency, and eventually saturates at a negative value at
the largestwc. However, in distinction to BCSI model,
∆ W (ωc) keeps varying with ωc up a much larger scale
and saturates only at around 0 .8eV . In between the dip
at 0 .1eV and 0 .8eV , the behavior of the optical integral is
predominantly determined by the variation of the cut-oﬀ
term ∆f(ωc) as evidenced by a close similarity between
the behavior of the actual ∆ W and ∆ W in the absence
  0.5 1  0   
0.1 
0.2 
Ω  in eV
σ ( Ω )
σ  in NS and SCS(EB model)
NS
SC
2 ∆ + ω o
ω o
0 0.2 0.4 0.6
190
200
λ  (the coupling constant)
W K  (meV)
Kubo Sum (EB model)
SC
NS
FIG. 7: Top- conductivities in the NS and the SCS for the EB
model. The conductivity in the NS vanishes belowω0 because
of no phase space for scattering. Bottom - Kubo sums as a
function of coupling. Observe thatWK in the SCS is below
that in the NS. We set ωo = 40 meV , ∆ = 30 meV , λ = .5
0 0.5 1
0.2
0.6
1  
ω c  in eV
W( ω c )/W( ∞ )
Normal State (EB model)
0 0.5 1
0.2
0.6
1  
ω c  in eV
W( ω c )/W( ∞ )
Superconducting State (EB model)
FIG. 8: Evolution of the optical integrals in the EB model.
Note thatW (0) has a non zero value at T = 0 in the NS
because the self-energy at small frequencies is purely real and
linear in ω, hence the polarization bubble Π(0) ̸= 0, as in an
ideal Fermi gas. Parameters are the same as in ﬁg. 7