2
in a given band is compensated by an appropriate change
of the spectral weight in other bands such that the total
spectral weight, integrated over all bands, is conserved,
as in Eq. (1). Still, non-conservation of the spectral
weight within a given band is an interesting phenomenon
as the degree of non-conservation is an indicator of rele-
vant energy scales in the problem. Indeed, when relevant
energy scales are much smaller than the Fermi energy,
i.e., changes in the conductivity are conﬁned to a near
vicinity of a Fermi surface (FS), one can expandεk near
kF as εk = vF (k − kF ) + ( k − kF )2/(2mB) + O(k − kF )3
and obtain ∇2
⃗kx
ε⃗k ≈ 1/mB [this approximation is equiv-
alent to approximating the density of states (DOS) by a
constant]. ThenWK becomes πne2/(2mB) which does
not depend on temperature. The scale of the tempera-
ture dependence ofWK is then an indicator how far in
energy the changes in conductivity extend when, e.g., a
system evolves from a normal metal to a superconductor.
Because relevant energy scales increase with the interac-
tion strength, the temperature dependence ofWK is also
an indirect indicator of whether a system is in a weak,
intermediate, or strong coupling regime.
In a conventional BCS superconductor the only rele-
vant scales are the superconducting gap ∆ and the impu-
rity scattering rate Γ. Both are generally much smaller
than the Fermi energy, so the optical integral should be
almostT -independent, i.e., the spectral weight lost in a
superconducting state at low frequencies because of gap
opening is completely recovered by the zero-frequencyδ-
function. In a clean limit, the weight which goes into
aδ−function is recovered within frequencies up to 4∆.
This is the essence of FGT sum rule 2,3. In a dirty limit,
this scale is larger, O(Γ), but still WK is T -independent
and there was no “violation of sum rule”.
The issue of sum rule attracted substantial interest in
the studies of high Tc cuprates5–18,21–26 in which pairing
is without doubts a strong coupling phenomenon. From a
theoretical perspective, the interest in this issue was orig-
inally triggered by a similarity betweenWK and the ki-
netic energy K = 2 ∑ ε⃗kn⃗k.18–20 For a model with a sim-
ple tight binding cosine dispersion εk ∝ (cos kx + cos ky),
d2 ε⃗k
d k2x
∼ −ε⃗k and WK = −K. For a more complex dis-
persion there is no exact relation between WK and K,
but several groups argued 17,27,28 that WK can still be
regarded as a good monitor for the changes in the kinetic
energy. Now, in a BCS superconductor, kinetic energy
increases belowTc because nk extends to higher frequen-
cies (see Fig.2). At strong coupling, K not necessary
increases because of opposite trend associated with the
fermionic self-energy: fermions are more mobile in the
SCS due to less space for scattering at low energies than
they are in the NS. Model calculations show that above
some coupling strength, the kinetic energy decreases be-
lowTc29. While, as we said, there is no one-to-one cor-
respondence between K and WK , it is still likely that,
when K decreases, WK increases.
A good amount of experimental eﬀort has been put into
addressing the issue of the optical sum rule in the c−axis7
and in-plane conductivities 8–16 in overdoped, optimally
doped, and underdoped cuprates. The experimental re-
sults demonstrated, above all, outstanding achievements
of experimental abilities as these groups managed to de-
tect the value of the optical integral with the accuracy
of a fraction of a percent. The analysis of the change
of the optical integral between normal and SCS is even
more complex because one has to (i) extend NS data to
T < Tc and (ii) measure superﬂuid density with the same
accuracy as the optical integral itself.
The analysis of the optical integral showed that in over-
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