text: τ z =  χ234/(√3/4) =  (4/√3)S2  (S3  S4)  (6) 
  
  − 
  The above representations of τ x,y,z are all invariant under 
  global spin rotation of the physical spins. 
  
  −  ·  × 
  
  With the machinery of equations (4), (5), and (6), it 
  will be straightforward to construct various pseudo-spin- 
  1/2 Hamiltonians on various lattices, of the Kitaev vari- 
  ety and beyond, as the exact low energy eﬀective Hamil- 
  tonian of certain spin-1/2 models with spin-rotation sym- 
  metry. In these constructions a pseudo-spin lattice site 
  actually represents a cluster of four spin-1/2 moments.  It is simpler to consider the permutation operators 
  Sk + 1/2, which just exchange the states 
  = k). 
  Pjk ≡ 
  of the two physical spin-1/2 moments j and k (j 
  As an example we consider the action of P34, 
  
  2Sj · 
  
  III. REALIZATION OF THE KITAEV MODEL. 
  + ω2  + ω 
  | ↓↑↑↓i  | ↓↑↓↑i 
  
  + ω2 
  | ↑↓↑↓i(cid:17) 