text: where n~k is the electronic distribution function and ε~k 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 
  m G(~k, ωm). For ǫk = k2/2m, 
  function as n~k = T 
  ε~k = 1/m, WK = πne2/(2m), and Kubo sum rule 
  ∇2 
  ~kx 
  reduces to Eq. (1). In general, however, ε~k 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 n~k. 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 with T . This issue was ﬁrst studied in detail by 
  Hirsch 4 who introduced the now-frequently-used nota- 
  tion “violation of the conductivity sum rule”. 
  
  P 