8 0.2 0.4 0.6 0.8 −40 0 $_{c}$ in eV W SC $_{(}$ω c ) − W NS $_{(}$ω c )  W (EB model) with lattice without lattice  W$_{K}$ 0.2 0.6 1 0 0.1 0.2  in eV $_{(}$ω) Conductivities (Original MFLI) NS SC + 1 0 50 100 120 130 140  (meV) Original MFLI W K  (meV) SC NS  =0.75 BCSI model. However, before that, we show in Figs 10- 12 the conductivities and the optical integrals for the original MFLI model. FIG. 9: Δ W vs the cut-off for the EB model. It remains neg- ative for larger cut-offs. Parameters are the same as before. The dot indicates the value of Δ W $^{(}$∞) = Δ W K of the lattice (the dashed line in Fig. 9). For their analysis of the optical integral, Norman and P´ epin 30 introduced a phenomenological model for the self energy which fits normal state scattering rate measure- ments by ARPES $^{41}$. It constructs the NS Σ $^{′′}$( ω ) out of two contributions - impurity scattering and electron- electron scattering which they approximated phenomeno- logically by the marginal Fermi liquid form of αω at small frequencies 6 (MFLI model). The total Σ ′′ is where ω$_{sat}$ is about ∼ 1 2 of the bandwidth, and f ( x ) ≈ 1 for x < 1 and decreases for x > 1. In Ref 30 f ( x ) was assumed to scale as 1 /x at large x such that Σ ′′ is flat at large ω . The real part of Σ( ω ) is obtained from Kramers- Kr¨ onig relations. For the superconducting state, they obtained Σ ′′ by cutting off the NS expression on the lower end at some frequency ω$_{1}$ (the analog of ω$_{0}$ + Δ that we had for EB model): We will show below that the sign of Δ W in the MFLI model actually depends on how the normal state results are extended to the superconducting state and, moreover, will argue that Δ W$_{K}$ is actually negative if the extension is done such that at α = 0 the results are consistent with The rationale for Δ W$_{K}$ > 0 in the original MFLI model has been provided in Ref. 30. They argued that this is closely linked to the absence of quasiparticle peaks in the NS and their restoration in the SCS state because the phase space for quasiparticle scattering at low ener- gies is smaller in a superconductor than in a normal state. In Fig 10 we plot the conductivities in the NS and the SCS and Kubo sums W$_{K}$ vs Γ at α = 0 . 75 showing that the spectral weight in the SCS is indeed larger than in the NS. In Fig 11 we show the behavior of the optical sums W ( ω$_{c}$ ) in NS and SCS. The observation here is that only ∼ 75 − 80% of the Kubo sum is recovered up to the scale of the bandwidth implying that there is indeed a significant spectral weight well beyond the bandwidth. And in Fig 12 we show the behavior of Δ W ( w$_{c}$ ). We see that it does not change sign and remain positive at all ω$_{c}$ , very much unlike the BCS case. Comparing the behavior of W ( w$_{c}$ ) with and without a lattice (solid and dashed lines in Fig. 12) we see that the ‘finite bandwidth effect’ just shifts the curve in the positive direction. We also see that the solid line flattens above roughly half of the bandwidth, i.e., at these frequencies Δ W ( ω$_{c}$ ) ≈ Δ W$_{K}$ . Still, we found that Δ W continues going down even above the bandwidth and truly saturates only at about 2 eV (not shown in the figure) supporting the idea that there is ‘more’ left to recover from higher frequencies. FIG. 10: Top –the conductivities in the NS and SCS in the original MFLI model of Ref.30. We set Γ = 70 meV , α = 0 . 75, Δ = 32 meV , ω 1 = 71 meV . Note that σ $^{′}$( ω ) in the SCS begins at Ω = Δ + ω 1 . Bottom – the behavior of W K with Γ. Σ ′′ ( ω ) = Γ + α | ω | f ( ω ω$_{sat}$ ) (17) Σ ′′ ( ω ) = (Γ + α | ω | )Θ( | ω | − ω$_{1}$ ) (18) where Θ( x ) is the step function. In reality, Σ ′′ which fits ARPES in the NS has some angular dependence along the Fermi surface $^{42}$, but this was ignored for simplicity. This model had gained a lot of attention as it predicted the optical sum in the SCS to be larger than in the NS, i.e., Δ W > 0 at large frequencies. This would be consistent with the experimental findings in Refs. 8,9 if, indeed, one identifies Δ W measured up to 1eV with Δ W$_{K}$ . C. Marginal Fermi liquid model