9
0 0.5 10
0.2
0.4
0.6
0.8
1
ω c  in eV
W( ω c )/W( ∞ ) 
NS (Original MFLI)
0 0.5 1
0.2
0.6
1  
ω c  in eV
W( ω c )/W( ∞ )
SCS (Original MFLI)
FIG. 11: The evolution of the optical integral in the NS (top)
and the SCS (bottom) in the original MFLI model. Parame-
ters are the same as above. Note that only∼ 75 − 80% of the
spectral weight is recovered up to 1 eV .
0.2 0.4 0.6 0.8 1
0 
10
20
ω c  in eV
W SC ( ω c ) − W NS ( ω c )
NS and SCS ∆ W (Original MFLI)
with lattice
without lattice
∆  W K
FIG. 12: Evolution of the diﬀerence of the optical integrals in
the SCS and the NS with the upper cut-oﬀ ωc. Parameters are
the same as before. Observe that the optical sum in the SCS
is larger than in the NS and that ∆W has not yet reached
∆ WK up to the bandwidth. The dashed line is the FGT
result.
This clearly aﬀectsnk because it is expressed via the full
Green’s function and competes with the conventional ef-
fect of the gap opening. The distribution function from
this model, which we show in Fig.2b brings this point
out by showing that in a MFLI model, atǫ < 0, nk in a
superconductor is larger than nk in the normal state, in
clear diﬀerence with the BCSI case.
We analyzed the original MFLI model for various pa-
rameters and found that the behavior presented in Fig.
12, where ∆W (ωc) > 0 for all frequencies, is typical but
0 20 40
175
185
195
Γ  in meV
W K  (meV)
Original MFLI in BCS limit
SC
NS
α  =0.05
FIG. 13: Behavior of WK with Γ for the original MFLI model
at very small α = 0 .05. We set ω1 = ∆ = 32 meV . Observe
the inconsistency with WK in the BCSI model in Fig 4.
0.2 0.4 0.6 0.8
−0.4
0
0.4 
ω c  in eV
W SC ( ω c ) − W NS ( ω c )
Original MFLI−two sign changes
FIG. 14: The special case of α = 1 .5,Γ = 5 meV , other pa-
rameters the same as in Fig. 10. These parameters are chosen
to illustrate that two sign changes (indicated by arrows in the
ﬁgure) are also possible within the original MFLI model.
not not a generic one. There exists a range of parame-
tersα and Γ where ∆ WK is still positive, but ∆ W (ωc)
changes the sign twice and is negative at intermediate
frequencies. We show an example of such behavior in
Fig14. Still, for most of the parameters, the behavior of
∆W (ωc) is the same as in Fig. 12.
On more careful looking we found the problem with the
original MFLI model. We recall that in this model the
self-energy in the SCS state was obtained by just cutting
the NS self energy atω1 (see Eq.18). We argue that
this phenomenological formalism is not fully consistent,
at least for smallα. Indeed, for α = 0, the MFLI model
reduces to BCSI model for which the behavior of the self-
energy is given by Eq. (12). This self-energy evolves with
ωand Σ
′′
has a square-root singularity at ω = ∆ + ωo
(with ωo = 0). Meanwhile Σ
′′
in the original MFLI model
in Eq. (18) simply jumps to zero at ω = ω1 = ∆, and
this happens for all values of α including α = 0 where the
MFLI and BCSI model should merge. This inconsistency
is reﬂected in Fig 13, where we plot the near-BCS limit
of MFLI model by taking a very smallα = 0 .05. We
see that the optical integral WK in the SCS still remains
larger than in the NS over a wide range of Γ, in clear
diﬀerence with the exactly known behavior in the BCSI