text: where Zo ∼ 2 ωsf χ0/ ∂Π 
  ∂ω 
  
  . The imaginary part 
  | 
  of the spin susceptibility describing a gaped continuum 
  exists for for Ω ≥ 2∆ and is 
  
  Ω=ω0 
  
  The collective nature of spin ﬂuctuations is reﬂected in 
  the fact that the coupling λ and the bosonic frequency 
  ωsf are related: λ scales as ξ2, where ξ is the bosonic 
  mass (the distance to a bosonic instability), and ωsf ∝ 
  2 (see Ref. 49). For a ﬂat χ(q ∼ Q) the product λωsf 
  ξ− 
  does not depend on ξ and is the overall dimensional scale 
  for boson-mediated interactions. 
  
  ′′ 
  χ  (q, Ω) = Im 
  1 − 1 
  ωsf 
  " 
  
  (cid:0) 
  
  χ0 
  π∆2 
  Ω + i π 
  ≈ Im 
  1 − 1 
  ωsf  2 Ω  "  # 
  
  (cid:1) 
  In Eq. (23) D(x) = K1(x) 
  K2(x) 
  , and K1(x) and K2(x) 
  are Elliptic integrals of ﬁrst and second kind. The real 
  part of χ is obtained by Kramers-Kr¨onig transform of the 
  imaginary part. 
  
  (cid:0) 
  − 
  x 