10
0 0.5 10  
0.4
0.8
Conductivities (Corrected MFLI)
σ ( ω )
ω  in eV
NS
SC
2 ∆
0 50 100
100
120
W K  (meV)
Γ  in meV
Corrected MFLI
SC
NS
FIG. 15: Top – σ(ω) in the NS and the SCS in the ‘corrected’
MFLI model with the feedback from SC on the quasiparticle
damping:iΓ term transforms into Γ
√
−ω 2+∆ 2 . In the SCS σ
now begins at Ω = 2∆. The parameters are same as in Fig.
10. Bottom – the behavior of Kubo sum with Γ. Observe
thatW (ωc) in the NS is larger than in the SCS.
0.2 0.4 0.6 0.8
−10
0  
10 
ω c  in eV
W SC ( ω c ) − W NS ( ω c )
Corrected MFLI
without lattice
with lattice
∆  W K
FIG. 16: Evolution of the diﬀerence of the optical integrals
between the SCS and the NS with the upper cut-oﬀωc for
the “corrected” MFLI model. Now ∆ W (ωc) is negative above
some frequency. Parameters are same as in the Fig 15.
model, whereWK is larger in the NS for all Γ (see Fig.
4). In other words, the original MFLI model does not
have the BCSI theory as its limiting case.
We modiﬁed the MFLI model is a minimal way by
changing the damping term in a SCS to Γ
√
−ω2+∆ 2 to be
consistent with BCSI model. We still use Eq. (18) for
the MFL term simply because this term was introduced
in the NS on phenomenological grounds and there is no
way to guess how it gets modiﬁed in the SCS state with-
out ﬁrst deriving the normal state self-energy microscop-
ically (this is what we will do in the next section). The
results of the calculations for the modiﬁed MFLI model
are presented in Figs. 15 and 16. We clearly see that the
behavior is now diﬀerent and ∆WK < 0 for all Γ. This
is the same behavior as we previously found in BCSI
and EB models. So we argue that the ‘unconventional’
behavior exhibited by the original MFLI model is most
likely the manifestation of a particular modeling incon-
sistency. Still, Ref. 30 made a valid point that the fact
that quasiparticles behave more close to free fermions in
a SCS than in a NS, and this eﬀect tends to reverse the
signs of ∆WK and of the kinetic energy 43. It just hap-
pens that in a modiﬁed MFLI model the optical integral
is still larger in the NS.
D. The collective boson model
We now turn to a more microscopic model- the CB
model. The model describes fermions interacting by ex-
changing soft, overdamped collective bosons in a partic-
ular, near-critical, spin or charge channel31,44,45. This
interaction is responsible for the normal state self-energy
and also gives rise to a superconductivity. A peculiar
feature of the CB model is that the propagator of a col-
lective boson changes belowTc because this boson is not
an independent degree of freedom (as in EB model) but
is made out of low-energy fermions which are aﬀected by
superconductivity32.
The most relevant point for our discussion is that this
model contains the physics which we identiﬁed above as
a source of a potential sign change of ∆WK . Namely,
at strong coupling the fermionic self-energy in the NS
is large because there exists strong scattering between
low-energy fermions mediated by low-energy collective
bosons. In the SCS, the density of low-energy fermions
drops and a continuum collective excitations becomes
gaped. Both eﬀects reduce fermionic damping and lead
to the increase ofWK in a SCS. If this increase exceeds a
conventional loss of WK due to a gap opening, the total
∆ WK may become positive.
The CB model has been applied numerous times to the
cuprates, most often under the assumption that near-
critical collective excitations are spin ﬂuctuations with
momenta nearQ = ( π, π). This version of a CB bo-
son is commonly known as a spin-fermion model. This
model yieldsdx2−y2 superconductivity and explains in a
quantitative way a number of measured electronic fea-
tures of the cuprates, in particular the near-absence of
the quasiparticle peak in the NS of optimally doped and
underdoped cuprates39 and the peak-dip-hump structure
in the ARPES proﬁle in the SCS 31,32,46,47. In our analy-
sis we assume that a CB is a spin ﬂuctuation.