text: We emphasize that because the ﬁrst intra-cluster term 
  Pcluster Hcluster commutes with the latter Kitaev terms 
  independent of the representation used, the Kitaev model 
  is realized as the exact low energy Hamiltonian of this 
  model without truncation errors of perturbation theories, 
  /Jcluster)2 or higher order terms will 
  namely no ( 
  | 
  be generated under the projection to low energy clus- 
  ter singlet space. This is unlike, for example, the t/U 
  expansion of the half-ﬁlled Hubbard model22,23, where 
  at lowest t2/U order the eﬀective Hamiltonian is the 
  Heisenberg model, but higher order terms (t4/U 3 etc.) 
  should in principle still be included in the low energy ef- 
  fective Hamiltonian for any ﬁnite t/U . Similar compari- 
  son can be made to the perturbative expansion studies of 
  the Kitaev-type models by Vidal et al.9, where the low 
  energy eﬀective Hamiltonians were obtained in certian 
  anisotropic (strong bond/triangle) limits. Although the 
  spirit of this work, namely projection to low energy sec- 
  tor, is the same as all previous perturbative approaches 
  to eﬀective Hamiltonians. 
  
  Jx,y,z| 