14
modiﬁed MFLI models. It is interesting that this holds
despite the fact that for largeλ CB model displays the
physics one apparently needs to reverse the sign of ∆ WK
– the absence of the quasiparticle peak in the NS and its
emergence in the SCS accompanied by the dip and the
hump at larger energies. The absence of coherent quasi-
particle in the NS at largeλ is also apparent form Fig
21 where we show the normal state distribution functions
for two diﬀerentλ. For large λ the jump (which indicates
the presence of quasiparticles) virtually disappears.
On a more careful look, we found that indiﬀerence of
δW (ωc) to the increase of λ is merely the consequence of
the fact that above we kept λωsf constant. Indeed, at
small frequencies, fermionic self-energy in the NS is Σ ′ =
λω, Σ” = λ2ω2/(λωsf ), and both Σ ′ and Σ ′′ increase
with λ if we keep λωsf constant. But at frequencies larger
than ωsf , which we actually probe by ∆ W (ωc), the self-
energy essentially depends only on λωsf , and increasing λ
but keeping λωsf constant does not bring us closer to the
physics associated with the recovery of electron coherence
in the SCS. To detect this physics, we need to see how
things evolve when we increaseλωsf above the scale of
∆ , i.e., consider a truly strong coupling when not only
λ≫ 1 but also the normal state Σ NS (ω ≥ ∆) >> ∆.
To address this issue, we took a larger λ for the same
ωsf and re-did the calculation of the conductivities and
optical integrals. The results for σ(ω) and ∆ W (ωc) are
presented in Fig. 22. We found the same behavior as be-
fore, i.e., ∆WK is negative. But we also found that the
larger is the overall scale for the self-energy, the larger is a
frequency of zero-crossing of ∆W (ωc). In particular, for
the same λ and ωsf that were used in Ref. 33 to ﬁt the NS
conductivity data, the zero crossing is at ∼ 0.8 eV which
is quite close to the bandwidth. This implies that at a
truly strong coupling the frequency at which ∆W (ωc)
changes sign can well be larger than the bandwidth of
1eV in which case ∆ W integrated up to the bandwidth
does indeed remain positive. Such behavior would be
consistent with Refs.8,9. we also see from Fig. 22 that
∆WK becomes small at a truly strong coupling, and over
a wide range of frequencies the behavior of ∆ W (ωc) is
predominantly governed by ∆ f(ωc), i.e. by the cut-oﬀ
term.50 The implication is that, to ﬁrst approximation,
∆ WK can be neglected and positive ∆ W (wc) integrated
to a frequency where it is still positive is almost compen-
sated by the integral over larger frequencies. This again
would be consistent with the experimental data in Refs.
8,9.
It is also instructive to understand the interplay be-
tween the behavior of ∆ W (ωc) and the behavior of the
diﬀerence of the kinetic energy between the SCS and the
NS,δKE . We computed the kinetic energy as a function
of λωsf and present the results in Fig. 23 for λ = 1 and
10. For a relatively weak λ = 1 the behavior is clearly
BCS like- δKE > 0 and increases with increasing λωsf .
However, at large λ = 10, we see that the kinetic energy
begin decreasing at large λωsf and eventually changes
sign. The behavior of δKE at a truly strong coupling is
consistent with earlier calculation of the kinetic energy
for Ornstein-Zernike form of the spin susceptibility43.
We clearly see that the increase of the zero crossing
frequency of ∆ W (ωc) at a truly strong coupling is cor-
related with the non-BCS behavior of δKE . At the same
time, the behavior of δW (ωc) is obviously not driven by
the kinetic energy as eventually δW (ωc) changes sign and
become negative. Rather, the increase in the frequency
range where ∆W (ωc) remains positive and non-BCS be-
havior of δKE are two indications of the same eﬀect that
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