or Kubo sum rule
∫ ‘∞′
0
Re σ(Ω) dΩ = WK = πe2
2N
∑
⃗k
∇2
⃗kx
ε⃗k n⃗k (3)
where n⃗k is the electronic distribution function and ε⃗k is
the band dispersion. Prime in the upper limit of the inte-
gration has the practical implication that the upper limit
is much larger than the bandwidth of a given band which
crosses the Fermi level, but smaller than the frequencies
of interband transitions. Interactions with external ob-
jects, e.g., phonons or impurities, and interactions be-
tween fermions are indirectly present in the distribution
function which is expressed via the full fermionic Green’s
function asn⃗k = T ∑
m G(⃗k, ωm). For ǫk = k2/2m,
∇2
⃗kx
ε⃗k = 1 /m, WK = πne2/(2m), and Kubo sum rule
reduces to Eq. (1). In general, however, ε⃗k is a lattice
dispersion, and Eqs. (1) and (3) are diﬀerent. Most im-
portant,WK in Eq. (3) generally depends on T and on
the state of the system because of n⃗k. In this situation,
the temperature evolution of the optical integral does not
reduce to a simple redistribution of the spectral weight
– the whole spectral weight inside the conduction band
changes withT . This issue was ﬁrst studied in detail by
Hirsch 4 who introduced the now-frequently-used nota-
tion “violation of the conductivity sum rule”.
In reality, as already pointed out by Hirsch, there is no
true violation as the change of the total spectral weight