4
an energy of interband transitions, which is roughly 2 eV .
This would be consistent with Refs. 8,9.
We begin with formulating our calculational basis in
the next section. Then we take up the four cases and
consider in each case the extent to which the Kubo sum is
satisﬁed up to the order of bandwidth and the functional
form and the sign of ∆W (ωc). The last section presents
our conclusions.
II. OPTICAL INTEGRAL IN NORMAL AND
SUPERCONDUCTING STATES
The generic formalism of the computation of the op-
tical conductivity and the optical integral has been dis-
cussed several times in the literature21–23,26,29 and we
just list the formulas that we used in our computations.
The conductivityσ(Ω) and the optical integral W (ωc)
are given by (see for example Ref. 35).
σ′(Ω) = Im
[
− Π(Ω)
Ω + iδ
]
= − Π ′′(Ω)
Ω + πδ(Ω) Π ′(Ω)
(7a)
W (ωc) =
∫ ωc
0
σ′(Ω) dΩ = −
∫ ωc
0+
Π ′′(Ω)
Ω dΩ + π
2 Π ′(0)
(7b)
where ‘X′’ and ‘X′′’ stand for real and imaginary parts
of X. We will restrict with T = 0. The polarization
operator Π(Ω) is (see Ref. 36)
Π( iΩ) = T
∑
ω
∑
⃗k
(∇⃗kε⃗k)2
(
G(iω, ⃗k)G(iω + iΩ ,⃗k) + F (iω, ⃗k)F (iω + iΩ ,⃗k)
)
(8a)
Π ′′(Ω) = − 1
π
∑
⃗k
(∇⃗kε⃗k)2
∫ 0
−Ω
dω
(
G′′(ω, ⃗k)G′′(ω + Ω ,⃗k) + F ′′(ω, ⃗k)F ′′(ω + Ω ,⃗k)
)
(8b)
Π ′(Ω) = 1
π2
∑
⃗k
(∇⃗kε⃗k)2
∫ ′ ∫ ′
dx dy
(
G′′(x,⃗k)G′′(y, ⃗k) + F ′′(x,⃗k)F ′′(y, ⃗k)
) nF (y) − nF (x)
y − x (8c)
where
∫′ denotes the principal value of the integral,∑
⃗k is understood to be 1
N
∑
⃗k,(N is the number of lat-
tice sites), nF (x) is the Fermi function which is a step
function at zero temperature, G and F are the normal
and anomalous Greens functions. given by 37
For a NS, G(ω, ⃗k) = 1
ω − Σ( k, ω) − ε⃗k + iδ (9a)
For a SCS, G(ω, ⃗k) = Zk,ωω + ε⃗k
Z2
k,ω (ω2 − ∆ 2
k,ω) − ε2
⃗k + iδsgn(ω)
(9b)
F (ω, ⃗k) = Zk,ω∆ k,ω
Z2
k,ω(ω2 − ∆ 2
k,ω) − ε2
⃗k + iδsgn(ω)
(9c)
where Zk,ω = 1 − Σ( k,ω)
ω , and ∆ k,ω, is the SC gap. Fol-
lowing earlier works 31,33, we assume that the fermionic
self-energy Σ( k, ω) predominantly depends on frequency
and approximate Σ( k, ω) ≈ Σ( ω) and also neglect the
frequency dependence of the gap, i.e., approximate ∆ k,ω
by a d−wave ∆ k. The lattice dispersion ε⃗k is taken from
Ref. 38. To calculate WK , one has to evaluate the Kubo
term in Eq.3 wherein the distribution function n⃗k, is cal-
culated from
n(ε⃗k) = −2
∫ 0
−∞
dω
2π G′′(ω, ⃗k) (10)
The 2 is due to the trace over spin indices. We show the
distribution functions in the NS and SCS under diﬀerent
circumstances in Fig 2.
The ⃗k-summation is done over ﬁrst Brillouin zone for a
2-D lattice with a 62x62 grid. The frequency integrals are
done analytically wherever possible, otherwise performed
using Simpson’s rule for all regular parts. Contributions
from the poles are computed separately using Cauchy’s
theorem. For comparison, in all four cases we also calcu-
lated FGT sum rule by replacing
∫
d2k = dΩ kdǫkνǫk,Ω k
and keeping ν constant. We remind that the FGT is
the result when one assumes that the integral in W (ωc)
predominantly comes from a narrow region around the
Fermi surface.
We will ﬁrst use Eq 3 and compute WK in NS and SCS.
This will tell us about the magnitude of ∆ W (ωc = ∞).
We next compute the conductivity σ(ω) using the equa-
tions listed above, ﬁnd W (ωc) and ∆ W (ωc) and compare
∆ f(ωc) and ∆ WK .
For simplicity and also for comparisons with earlier
studies, for BCSI, EB, and MFLI models we assumed
that the gap is just a constant along the FS. For CB
model, we used ad−wave gap and included into consid-
eration the fact that, if a CB is a spin ﬂuctuation, its
propagator develops a resonance when the pairing gap is
d−wave.