text: (7) 
  
  Many other representations are possible as well, because 
  several physical spin interactions may correspond to the 
  same pseudo-spin interaction in the physical singlet sec- 
  tor, and we will take advantage of this later. 
  
  where j, k label the honeycomb lattice sites thus the four- 
  spin clusters, Hcluster is given by (2), τ x,y,z should be 
  replaced by the corresponding physical spin operators in 
  (4) and (5) or (6), or some other equivalent representa- 
  tions of personal preference. 
  
  For τ z we can use τ z =  iτ xτ y, where i is the imagi- 
  − 
  nary unit, 
  
  i(2/√3)(2S1  S2 + 1/2)S1  (S3  S4) 
  −  ·  ·  − 