5
FIG. 2: Distribution functions in four cases (a) BCSI model,
where one can see that forε > 0, SC >NS implying KE in-
creases in the SCS. (b) The original MFLI model of Ref. 30,
where forε > 0, SC <NS, implying KE decreases in the SCS.
(c) Our version of MFLI model (see text) and (d) the CB
model. In both cases, SC>NS, implying KE increases in the
SCS. Observe that in the impurity-free CB model there is no
jump inn(ǫ) indicating lack of fermionic coherence. This is
consistent with ARPES 39
A. The BCS case
In BCS theory the quantity Z(ω) is given by
ZBCSI (ω) = 1 + Γ
√
∆ 2 − (ω + iδ)2 (11)
and
Σ BCSI (ω) = ω (Z(ω) − 1) = iΓ ω√
(ω + iδ)2 − ∆ 2 (12)
This is consistent with having in the NS, Σ = iΓ in accor-
dance with Eq 6. In the SCS, Σ( ω) is purely imaginary
for ω > ∆ and purely real for ω < ∆. The self-energy
has a square-root singularity at ω = ∆.
It is worth noting that Eq.12 is derived from the in-
tegration over inﬁnite band. If one uses Eq.6 for ﬁnite
band, Eq.12 acquires an additional frequency dependence
at large frequencies of the order of bandwidth (the low
frequency structure still remains the same as in Eq.12).
In principle, in a fully self-consistent analysis, one should
indeed evaluate the self-energy using a ﬁnite bandwidth.
In practice, however, the self-energy at frequencies of or-
der bandwidth is generally much smaller thanω and con-
tribute very little to optical conductivity which predom-
inantly comes from frequencies where the self-energy is
comparable or even larger thanω. Keeping this in mind,
below we will continue with the form of self-energy de-
rived form inﬁnite band. We use the same argument for
all four models for the self-energy.
For completeness, we ﬁrst present some well known
results about the conductivity and optical integral for a
constant DOS and then extend the discussion to the case
where the same calculations are done in the presence of
a particular lattice dispersion.
0.1 0.2 0.3 0.4
−4
−2
0 
ω  in eV
W SC −W NS
∆  W (BCSI without lattice)
Γ  =70 meV
Γ  =50 meV
Γ  =3.5 meV
FIG. 3: The BCSI case with a dispersion linearized around the
Fermi surface. Evolution of the diﬀerence of optical integrals
in the SCS and the NS with the upper cut-oﬀ ωc Observe
that the zero crossing point increases with impurity scatte ring
rate Γ and also the ‘dip’ spreads out with increasing Γ. ∆ =
30meV
For a constant DOS, ∆ W (ωc) = WSC (ωc) − WNS (ωc)
is zero at ωc = ∞ and Kubo sum rule reduces to FGT
sum rule. In Fig. 3 we plot for this case ∆ W (ωc) as a
function of the cutoﬀ ωc for diﬀerent Γ ′s. The plot shows
the two well known features: zero-crossing point is below
2∆ in the clean limit Γ<< ∆ and is roughly 2Γ in the
dirty limit21,40 The magnitude of the ‘dip’ decreases quite
rapidly with increasing Γ. Still, there is always a point
of zero crossing and ∆W (ωc) at large ωc approaches zero
from below.
We now perform the same calculations in the presence
of lattice dispersion. The results are summarized in Figs
4,5, and 6.
Fig 4 shows conductivities σ(ω) in the NS and the SCS
and Kubo sums WK plotted against impurity scattering
Γ. We see that the optical integral in the NS is always
greater than in the SCS. The negative sign of ∆WK is
simply the consequence of the fact that nk is larger in the
NS for ǫk < 0 and smaller for ǫk < 0, and ∇2ε⃗k closely
follows −ε⃗k for our choice of dispersion 38), Hence nk is
larger in the NS for ∇2ε⃗k > 0 and smaller for ∇2ε⃗k <
0 and the Kubo sum rule, which is the integral of the
product ofnk and ∇2ε⃗k (Eq. 3), is larger in the normal
state.
We also see from Fig. 4 that ∆ WK decreases with Γ
reﬂecting the fact that with too much impurity scattering
there is little diﬀerence innk between NS and SCS.
Fig 5 shows the optical sum in NS and SCS in clean
and dirty limits (the parameters are stated in the ﬁg-
ure). This plot shows that the Kubo sums are almost
completely recovered by integrating up to the bandwidth
of 1eV : the recovery is 95% in the clean limit and ∼ 90%