text: 1 
  ) 
  
  ∞ 
  W 
  ( 
  
  0.6 
  / 
  ) 
  c 
  
  ω 
  W 
  ( 
  
  0.2 
  
  0 
  0.5 
  ω 
   in eV 
  c 
  
  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 
  80% of the 
  spectral weight is recovered up to 1eV . 
  75 
  ∼  − 
  
  NS and SCS ∆W (Original MFLI) 
  
  with lattice 
  without lattice 
  
  ) 
  c 
  20 
  ω 
  ( 
  
  S 
  N 
  W 
  − 
  ) 
  10 
  
  not not a generic one. There exists a range of parame- 
  ters α 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. 
  
  c 
  ω 
  ( 
  
  C 
  S 
  W 
  0 
  0.2 
  0.6 
  0.4 
  ω 
   in eV 
  c 
  
  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. 