fermions are incoherent in the NS but acquire coherence
in the SCS.
III. CONCLUSION
In this work we analyzed the behavior of optical in-
tegrals W (ωc) ∝
∫ωc
o σ(ω)dω and Kubo sum rules in
the normal and superconducting states of interacting
fermionic systems on a lattice. Our key goal was to
understand what sets the sign of ∆WK = ∆ W (∞) be-
tween the normal and superconducting states and what
is the behavior ofW (ωc) and ∆ W (ωc) at ﬁnite ωc. In a
weak coupling BCS superconductor, ∆ W (ωc) is positive
at ωc < 2∆ due to a contribution from superﬂuid den-
sity, but becomes negative at larger ωc, and approach a
negative value of ∆ WK . Our study was motivated by fas-
cinating optical experiments on the cuprates 7–10 . In over-
doped cuprates, there is clear indication 11 that ∆ W (ωc)
becomes negative above a few ∆, consistent with BCS
behavior. In underdoped cuprates, two groups argued8,9
that ∆ W integrated up to the bandwidth remains posi-
tive, while the other group argued 10 that it is negative.
The reasoning why ∆ WK may potentially change sign
at strong coupling involves the correlation between −WK
and the kinetic energy. In the BCS limit, kinetic en-
ergy obviously increases in a SCS because of gap opening,
hence−WK increases, and ∆ WK is negative. At strong
coupling, there is a counter eﬀect – fermions become more
mobile in a SCS due to a smaller self-energy.
We considered four models: a BCS model with impu-
rities, a model of fermions interacting with an Einstein
boson, a phenomenological MFL model with impurities,
and a model of fermions interacting with collective spin
ﬂuctuations. In all cases, we found that ∆WK is neg-
ative, but how it evolves with ωc and how much of the
sum rule is recovered by integrating up to the bandwidth
depends on the model.
The result most relevant to the experiments on the
cuprates is obtained for the spin ﬂuctuation model.
We found that at strong coupling, the zero-crossing of
δW(ωc) occurs at a frequency which increases with the
coupling strength and may become larger than the band-
width at a truly strong coupling. Still, at even larger
frequencies, ∆W (ωc) is negative.