text: ′ denotes the principal value of the integral, 
  ~k,(N is the number of lat- 
  tice sites), nF (x) is the Fermi function which is a step 
  P 
  P 
  function at zero temperature, G and F are the normal 
  and anomalous Greens functions. given by37 
  
  where 
  ~k is understood to be 1 
  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. 
  N 
  R 
  
  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- 
  d2k = dΩkdǫkνǫk,Ωk 
  lated FGT sum rule by replacing 
  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. 
  
  1 
  ω − Σ(k, ω) − ε~k + iδ 
  Zk,ωω + ε~k 
  k,ω) − ε2 
  ~k 
  + iδsgn(ω) 
  (9b) 
  
  Zk,ω∆k,ω 
  k,ω) − ε2 
  ~k 