7 FIG. 6: Evolution of Δ 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. Δ = 30 meV , Γ = 3 . 5 meV (clean limit), Γ = 150 meV (dirty limit) FIG. 7: Top- conductivities in the NS and the SCS for the EB model. The conductivity in the NS vanishes below ω 0 because of no phase space for scattering. Bottom - Kubo sums as a function of coupling. Observe that W K in the SCS is below that in the NS. We set ω o = 40 meV , Δ = 30 meV , λ = . 5 Observe that Σ ′′ ( ω ) is no-zero only for ω < − ω$_{o}$ − Δ. 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 W$_{K}$ 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 Δ W$_{K}$ 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 1 eV . The rest comes from higher frequencies. In Fig 9 we plot Δ W ( ω$_{c}$ ) as a function of ω$_{c}$ . We see the same behavior as in the BCSI model in a clean limit – Δ W ( ω$_{c}$ ) is positive at small frequencies, crosses zero at some ω$_{c}$ , passes through a deep minimum at a larger frequency, and eventually saturates at a negative value at the largest w$_{c}$ . However, in distinction to BCSI model, Δ W ( ω$_{c}$ ) keeps varying with ω$_{c}$ up a much larger scale and saturates only at around 0 . 8 eV . In between the dip at 0 . 1 eV and 0 . 8 eV , the behavior of the optical integral is predominantly determined by the variation of the cut-off term Δ f ( ω$_{c}$ ) as evidenced by a close similarity between the behavior of the actual Δ W and Δ W in the absence FIG. 8: Evolution of the optical integrals in the EB model. Note that W (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 $_{Σ}$′$_{(}$$_{ω}$$_{) =}$ − 1 2 λ$_{n}$ω$_{o}$ Re ∫ $_{dω}$′ 1 ω 2 o − ω ′ 2 − iδ ω + ω ′ √ ( ω + ω ′ ) 2 − Δ 2 (16) 0 0.1 0.2 0.3 0  20 40 $_{c}$ in eV W SC ( c ) − W NS ( c )  W (BCSI−clean limit)  with lattice without lattice 0.1 0.3 0.5 0  10 $_{c}$ in eV W SC ( c ) − W NS ( c )  W (BCSI−dirty limit) 0.5 1 −1 0 $_{c}$ in eV  W Larger  c 0.5 1 0 0.1  0.2   in eV ( )  in NS and SCS(EB model) NS SC 2 + o o 0 0.2 0.4 0.6 190 200  (the coupling constant) W K  (meV) Kubo Sum (EB model) SC NS 0 0.5 1 0.2 0.6 1 $_{c}$ in eV W( c )/W( ) Normal State (EB model) 0 0.5 1 0.2 0.6 1 $_{c}$ in eV W( c )/W( ) Superconducting State (EB model)