text: 6Jcluster PjSj2 
  Use (S)2 = 3/4 and 
  
  − 
  
  (Sk3 
  
  Sk4) 
  × 
  k3Sb 
  (Sa  (Sk3 
  k3Sb  Sk4) 
  k3Sb 
  Sa  · 
  k4Sa 
  × 
  k4 − 
  
  with λx = 
  Jcluster, and r = sgn(Jx) is the sign 
  Jx| · 
  | 
  of Jx. The non-trivial terms produced by up to second 
  order perturbation will be the τ x 
  k term. Note that the 
  last term in the above equation commutes with cluster 
  Hamiltonians so it does not produce second or higher 
  order perturbations. 
  
  k4Sb  k3Sa 
  = 
  k4) 
  X 
  a,b 
  
  k3Sb 
  Sa  k4Sb 
  Sa 
  = (Sk3  Sk3)(Sk4  Sk4) 
  k3[δab/2  k4] 
  
  − X 
  a,b 
  Sk4) 
  
  ·  ·  − 
  
  = 9/16 + (Sk3  Sk4)(Sk3  (3/8) 