text: However there is another simpler representation of τ z, 
  Sℓ). Ex- 
  by the spin-chirality operator χjkℓ = Sj · 
  S4) is 
  plicit calculation shows that the eﬀect of S2 
  (√3/4)τ z in the physical singlet sector. This can also 
  − 
  S3, S2 
  be proved by using the commutation relation [S2 
  · 
  S4). A complete list of all chirality 
  S4] = iS2 
  operators is given in TABLE I. Therefore we can choose 
  another representation of τ z, 
  
  (Sk × 
  (S3 
  ·  × 
  
  · 
  
  We now consider the eﬀect of Heisenberg-type inter- 
  Sk inside the physical singlet sector. Note 
  actions Sj · 
  Sk within the cluster commutes with 
  that since any Sj · 
  the cluster Hamiltonian Hcluster (2), their action do not 
  mix physical spin singlet states with states of other total 
  physical spin. This property is also true for the spin- 
  chirality operator used later. So the pseudo-spin Hamil- 
  tonian constructed below will be exact low energy Hamil- 
  tonian, without truncation errors in typical perturbation 
  series expansions. 