8
0.2 0.4 0.6 0.8
−40
0
ω c  in eV
W SC ( ω c ) − W NS ( ω c )
∆  W (EB model)
with lattice
without lattice
∆  W K
FIG. 9: ∆ 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 ∆W (∞) = ∆ WK
of the lattice (the dashed line in Fig. 9).
C. Marginal Fermi liquid model
For their analysis of the optical integral, Norman and
P´ epin30 introduced a phenomenological model for the self
energy which ﬁts normal state scattering rate measure-
ments by ARPES41. 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 ofαω at small
frequencies6 (MFLI model). The total Σ
′′
is
Σ ′′(ω) = Γ + α|ω|f
( ω
ωsat
)
(17)
where ωsat 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 ω1 (the analog of ω0 + ∆ that we
had for EB model):
Σ ′′(ω) = (Γ + α|ω|)Θ( |ω| − ω1) (18)
where Θ( x) is the step function. In reality, Σ
′′
which ﬁts
ARPES in the NS has some angular dependence along the
Fermi surface42, 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.,
∆W > 0 at large frequencies. This would be consistent
with the experimental ﬁndings in Refs. 8,9 if, indeed, one
identiﬁes ∆W measured up to 1eV with ∆ WK .
We will show below that the sign of ∆ W in the MFLI
model actually depends on how the normal state results
are extended to the superconducting state and, moreover,
will argue that ∆WK is actually negative if the extension
is done such that at α = 0 the results are consistent with
BCSI model. However, before that, we show in Figs 10-
12 the conductivities and the optical integrals for the
original MFLI model.
0.2 0.6 1
0
0.1
0.2
ω  in eV
σ ( ω )
Conductivities (Original MFLI)
NS
SC
∆ + ω 1
0 50 100120
130
140
Γ  (meV)
Original MFLI
W K  (meV)
SC
NS
α  =0.75
FIG. 10: Top –the conductivities in the NS and SCS in the
original MFLI model of Ref.30. We set Γ = 70meV , α = 0 .75,
∆ = 32 meV , ω1 = 71 meV . Note that σ′(ω) in the SCS
begins at Ω = ∆ + ω1. Bottom – the behavior of WK with Γ.
In Fig 10 we plot the conductivities in the NS and the
SCS and Kubo sums WK 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(ωc) 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 ∆W (wc). We see that it does
not change sign and remain positive at all ωc, very much
unlike the BCS case. Comparing the behavior of W (wc)
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 ∆W (ωc) ≈ ∆ WK . Still, we found that
∆ 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.
The rationale for ∆ WK > 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.