doped cuprates it deﬁnitely decreases below Tc, in con-
sistency with the expectations at weak coupling 11. For
underdoped cuprates, all experimental groups agree that
a relative change of the optical integral belowTc gets
much smaller. There is no agreement yet about the sign
of the change of the optical integral : Molegraafet al.8
and Santander-Syro et al.9 argued that the optical inte-
gral increases below Tc, while Boris et al.10 argued that
it decreases.
Theoretical analysis of these results 21,22,25,28,30 added
one more degree of complexity to the issue. It is tempt-
ing to analyze the temperature dependence ofWK and
relate it to the observed behavior of the optical integral,
and some earlier works25,28,30 followed this route. In the
experiments, however, optical conductivity is integrated
only up to a certain frequencyωc, and the quantity which
is actually measured is
W (ωc) =
∫ ωc
0
Re σ(Ω) dΩ = WK + f(ωc)
f(ωc) = −
∫ ′∞′
ωc
Re σ(Ω) dΩ (4)
The Kubo formula, Eq. (3) is obtained assuming that
the second part is negligible. This is not guaranteed,
however, as typicalωc ∼ 1 − 2eV are comparable to the
bandwidth.
The diﬀerential sum rule ∆ W is also a sum of two
terms
∆ W (ωc) = ∆ WK + ∆ f(ωc) (5)
where ∆ WK is the variation of the r.h.s. of Eq. 3,
and ∆ f(ωc) is the variation of the cutoﬀ term. Because
conductivity changes with T at all frequencies, ∆ f(ωc)
also varies with temperature. It then becomes the issue
whether the experimentally observed ∆W (ωc) is predom-
inantly due to “intrinsic” ∆ WK , or to ∆ f(ωc). [A third
possibility is non-applicability of the Kubo formula be-
cause of the close proximity of other bands, but we will
not dwell on this.]
For the NS, previous works 21,22 on particular models
for the cuprates indicated that the origin of the temper-
ature dependence ofW (ωc) is likely the T dependence
of the cutoﬀ term f(ωc). Speciﬁcally, Norman et. al.22
approximated a fermionic DOS by a constant (in which