3
case, as we said, WK does not depend on temperature)
and analyzed the T dependence of W (ωc) due to the T
dependence of the cut-oﬀ term. They found a good agree-
ment with the experiments. This still does not solve the
problem fully as amount of theT dependence of WK in
the same model but with a lattice dispersion has not been
analyzed. For a superconductor, which of the two terms
contributes more, remains an open issue. At small fre-
quencies, ∆W (ωc) between a SCS and a NS is positive
simply because σ(Ω) in a SCS has a δ−functional term.
In the models with a constant DOS, for which ∆ WK = 0,
previous calculations 21 show that ∆ W (ωc) changes sign
at some ωc, becomes negative at larger ωc and approaches
zero from a negative side. The frequency when ∆ W (ωc)
changes sign is of order ∆ at weak coupling, but increases
as the coupling increases, and at large coupling becomes
comparable to a bandwidth (∼ 1eV ). At such frequencies
the approximation of a DOS by a constant is question-
able at best, and the behavior of ∆W (ωc) should gen-
erally be inﬂuenced by a nonzero ∆ WK . In particular,
the optical integral can either remain positive for all fre-
quencies below interband transitions (for large enough
positive ∆WK ), or change sign and remain negative (for
negative ∆ WK ). The ﬁrst behavior would be consistent
with Refs. 8,9, while the second would be consistent with
Ref. 10. ∆W can even show more exotic behavior with
more than one sign change (for a small positive ∆ WK ).
We show various cases schematically in Fig.1.
0
ω c
∆  W
0
ω c
∆  W
0
ω c
∆  W 0
ω c
∆  W
(a)(a)(a) (b)
(c) (d)
FIG. 1: Schematic behavior of ∆ W vs ωc, Eq. (4). The
limiting value of ∆ W at ωc = ∞ is ∆ WK given by Eq. (3)
Depending on the value of ∆ WK , there can be either one sign
change of ∆ W (panels a and c), or no sign changes (panel b),
or two sign changes (panel d).
In our work, we perform direct numerical calculations
of optical integrals at T = 0 for a lattice dispersion ex-
tracted from ARPES of the cuprates. The goal of our
work is two-fold. First, we perform calculations of the
optical integral in the NS and analyze how rapidlyW (ωc)
approaches WK , in other words we check how much of
the Kubo sum is recovered up to the scale of the band-
width. Second, we analyze the diﬀerence between optical
integral in the SCS at T = 0 and in the NS extrapolated
to T = 0 and compare the cut oﬀ eﬀect ∆ f(ωc) to ∆ WK
term. We also analyze the sign of ∆ W (ωc) at large fre-
quencies and discuss under what conditions theoretical
W(∞) increases in the SCS.
We perform calculations for four models. First is a
conventional BCS model with impurities (BCSI model).
Second is an Einstein boson (EB) model of fermions in-
teracting with a single Einstein boson whose propaga-
tor does not change between NS and SCS. These two
cases will illustrate a conventional idea of the spectral
weight in SCS being less than in NS. Then we con-
sider two more sophisticated models: a phenomenological
“marginal Fermi liquid with impurities” (MFLI) model
of Norman and P´ epin30, and a microscopic collective bo-
son (CB) model 31 in which in the NS fermions interact
with a gapless continuum of bosonic excitations, but in a
d−wave SCS a gapless continuum splits into a resonance
and a gaped continuum. This model describes, in par-
ticular, interaction of fermions with their own collective
spin ﬂuctuations32 via
Σ( k, Ω) = 3 g2
∫ dω
2π
d2q
(2π)2 χ(q, ω)G(k + q, ω + Ω) (6)
where g is the spin-fermion coupling, and χ(q, ω) is the
spin susceptibility whose dynamics changes between NS
and SCS.
From our analysis we found that the introduction of
a ﬁnite fermionic bandwidth by means of a lattice has
generally a notable eﬀect on bothW and ∆ W . We
found that for all models except for BCSI model, only
70%− 80% of the optical spectral weight is obtained by
integrating up to the bandwidth. In these three models,
there also exists a wide range ofωc in which the behavior