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
of ∆ W (ωc) is due to variation of ∆ f(ωc) which is domi-
nant comparable to the ∆ WK term. This dominance of
the cut oﬀ term is consistent with the analysis in Refs.
21,22,33.
We also found that for all models except for the origi-
nal version of the MFLI model the optical weight at the
highest frequencies is greater in the NS than in the SCS
(i.e., ∆W < 0). This observation is consistent with the
ﬁndings of Abanov and Chubukov 32, Benfatto et. al.28,
and Karakozov and Maksimov 34. In the original ver-
sion of the MFLI model 30 the spectral weight in SCS
was found to be greater than in the NS (∆ W > 0). We
show that the behavior of ∆ W (ωc) in this model cru-
cially depends on how the fermionic self-energy modeled
to ﬁt ARPES data in a NS is modiﬁed when a system
becomes a superconductor and can be of either sign. We
also found, however, thatωc at which ∆ W becomes neg-
ative rapidly increases with the coupling strength and at
strong coupling becomes comparable to the bandwidth.
In the CB model, which, we believe, is most appropriate
for the application to the cuprates, ∆WK = ∆ W (∞) is
quite small, and at strong coupling a negative ∆ W (ωc)
up to ωc ∼ 1eV is nearly compensated by the optical
integral between ωc and “inﬁnity”, which, in practice, is