text: 1 
  √6 (cid:16)| ↓↓↑↑i 
  τ z =  P34  1  = 
  |  −  i 
  
  +  + ω 
  | ↑↑↓↓i  | ↑↓↓↑i 
  τ z = +1  = 
  
  | 
  τ z = 
  
  i 
  
  τ z = +1 
  
  and similarly P34 
  . Therefore P34 
  = 
  i 
  i 
  is just τ x in the physical singlet sector. A complete list 
  of all permutation operators is given in TABLE I. We 
  can choose the following representation of τ x and τ y, 
  
  1 
  |  −  | 
  
  the desired Hamiltonian in short notation, 
  
  τ x = P12 = 2S1 
  τ y = (P13 
  
  S2 + 1/2 
  · 
  (4) 
  Jxτ x 
  j τ x 
  k 
  H =  Hcluster 
  P14)/√3 = (2/√3)S1  (S3  S4) 
  X 
  cluster 
  − X 
  −  ·  − 
  
  x−links <jk> 
  Jyτ y 
  j τ y 
  Jzτ z 
  j τ z 
  k 
  k − X  − X 
  y−links <jk>  z−links <jk> 