13
00
0.5
1
ε
n( ε )
NS
SC Jump
λ =1
0   0
0.5
1
ε
n( ε )
NS
SC λ  =7
FIG. 21: Distribution functions n(ǫ) for CB model for λ = 1
and λ = 7 and a constant ωsf = 26 meV . We set ∆ = 30 meV .
For smaller λ (top), quasiparticles near the FS are well deﬁned
as indicated by the well pronounced jump in n(ǫ). For λ = 7,
n(ǫ) is rather smooth implying that a coherence is almost lost.
Some irregularities is the SCS distribution function are du e
to ﬁnite sampling in the frequency domain. The irregulariti es
disappear when ﬁner mesh for frequencies is chosen.
shows up in the optical gap), where as in the BCSI case
it would have always begun from 2∆. In Fig 18 we plot
the Kubo sumsWK vs coupling λ. We see that for all λ,
WK in the NS stays larger than in the SCS. Fig 19 shows
the cutoﬀ dependence of the optical integrals W (ωc) for
λ = 1 separately in the NS and the SCS. We again see
that only about 73% of the Kubo sum is recovered up
to the bandwidth of 1eV indicating that there is a sig-
niﬁcant amount left to recover beyond this energy scale.
Fig 20 shows ∆W for the two diﬀerent couplings. We
see that, for both λ’s, there is only one zero-crossing for
the ∆ W curve, and ∆ W is negative at larger frequen-
cies. The only diﬀerence between the two plots is that
for larger coupling the dip in ∆W gets ‘shallower’. Ob-
serve also that the solid line in Fig. 20 is rather far away
from the dashed line atωc > 1meV , which indicates that,
although ∆ W (ωc) in this region has some dependence on
ωc, still the largest part of ∆ W (ωc) is ∆ WK , while the
contribution from ∆ f(ωc) is smaller.
0 0.5 1
0.1
0.2
ω  in eV
σ ( ω )
Conductivities (CB model−larger λ  ω sf )
NS
SC
2 ∆  + ω o
0.2 0.4 0.6 0.8
0 
 
4 
8 
ω c  in eV
W SC ( ω c ) − W NS ( ω c )  in meV
∆  W (CB model−larger λ  ω sf )
with lattice
without lattice
FIG. 22: Top – conductivity at a larger value of ωsf λ (ωsf =
26 meV ,λ = 7) consistent with the one used in Ref.33). Bot-
tom – ∆ W with and without lattice. Observe that the fre-
quency of zero crossing of ∆ W enhances compared to the case
of a smaller λωsf and becomes comparable to the bandwidth.
At energies smaller than the bandwidth, ∆ W > 0, as in the
Norman- P´ epin model.
0 2 4 6
10
30
50
Self Energy prefactor
δ KE  in meV
λ =10
λ =1
FIG. 23: Kinetic energy diﬀerence between the SCS and the
NS,δKE We set λ to be either λ = 1 or λ = 10 and varied ωsf
thus changing the overall prefactor in the self-energy. At w eak
coupling ( λ = 1) the behavior is BCS-like – δKE is positive
and increases with the overall factor in the self-energy. At
strong coupling (λ = 7), δKE shows a reverse trend at larger
ωsf .
The negative sign of ∆ W (ωc) above a relatively small
ωc ∼ 0.1 − 0.2eV implies that the ‘compensating’ ef-
fect from the fermionic self-energy on ∆ W is not strong
enough to overshadow the decrease of the optical inte-
gral in the SCS due to gap opening. In other words,the
CB model displays the same behavior as BCSI, EB, and