12
form of the imaginary part.
0 0.5 10  
0.2
0.4
ω  in eV
σ ( ω )
Conductivities (CB model λ  =1)
NS
SC
2 ∆ + ω o
0 0.5 1
0.2
0.6
1  
ω  in eV
σ ( ω )
Conductivities (CB model λ  =10)
NS
SC
2 ∆  + ω o
FIG. 17: Conductivities and ∆ W for a ﬁxed λωsf . Top –
ωsf = 26 meV ,λ = 1, ωo = 40 meV ,Zo = 0 .77 Bottom –
ωsf = 2 .6 meV ,λ = 10, ωo = 13 .5 meV ,Zo = 1 .22. The zero
crossing for ∆ W is not aﬀected by a change in λ because it
is determined only by λωsf . We set ∆ = 30 meV .
1 2 3
120
160
200
λ  (coupling)
CB model ( Ω o =40 meV)
W K (meV)
SCS
NS
FIG. 18: The behavior of Kubo sums in the CB model. Note
that the spectral weight in the NS is always larger than in the
SCS. We setωsf = 26 meV ,λ = 1, and ∆ = 30 meV .
We performed the same calculations of conductivities
and optical integrals as in the previous three cases. The
results are summarized in Figs. 17 - 22. Fig 17 shows con-
ductivities in the NS and the SCS for two couplingsλ = 1
and λ = 10 (keeping λωsf constant). Other parameters
Zo and ωo are calculated according to the discussion after
Eq 21. for ωsf = 26 meV , λ = 1, we ﬁnd ωo = 40 meV ,
Zo = 0 .77. And for ωsf = 2 .6 meV , λ = 10, we ﬁnd
ωo = 13 .5 meV , Zo = 1 .22. Note that the conductivity
in the SCS starts at 2∆ + ωo (i.e. the resonance energy
0 0.5 1
0.2
0.6
1  
ω c  in eV
W( ω c )/W( ∞ )
NS Optical Sums (CB model)
0 0.5 1
0.2
0.6
1  
ω c  in eV
W( ω c )/W( ∞ )
SCS Optical Sums (CB model)
FIG. 19: The evolution of the optical integrals in the NS
and the SCS in the CB model. Note that about∼ 75% of
the spectral weight is recovered up to 1 eV . We set ωsf =
26 meV ,λ = 1, and ∆ = 30 meV .
0 0.5 1
−20
0  
20 
ω c  in eV
W SC ( ω c ) − W NS ( ω c )
∆  W (CB model λ =1)
with lattice
without lattice
∆  W K
0.2 0.6 1  
−20
0  
20 
ω c  in eV
W SC ( ω c ) − W NS ( ω c )
∆  W (CB model λ =10)
with lattice
without lattice
∆  W K
FIG. 20: ∆ W (in meV) for λ = 1(top) and λ = 10(bottom).
We used ωsf = 26 meV /λ and ∆ = 30 meV . The zero crossing
is not aﬀected because we keep λωsf constant. The notable
diﬀerence is the widening of the dip at a larger λ.