text: In this Section we will construct the pseudo-spin-1/2 
  from a cluster of four physical spins, and map the phys- 
  ical spin operators to pseudo-spin operators. The map- 
  ping constructed here will be used in later Sections to 
  construct the eﬀective Kitaev model. In this Section we 
  will work entirely within the four-spin cluster, all unspec- 
  iﬁed physical spin subscripts take values 1, . . . , 4. 
  
  Sk [or the spin-chirality Sj ·  (Sk × 
  
  Consider a cluster of four spin-1/2 moments(called 
  physical spins hereafter), 
  labeled by S1,...,4, antiferro- 
  magnetically coupled to each other (see the right bot- 
  tom part of FIG. 2). The Hamiltonian within the clus- 
  ter(up to a constant) is simply the Heisenberg antiferro- 
  magnetic(AFM) interactions, 
  
  One major drawback of the model to be presented is 
  that it contains high order spin interactions(involves up 
  to six or eight spins), thus is still unnatural. However it 
  opens the possibility to realize exotic (exactly solvable) 
  models from spin-1/2 Hamiltonian with spin rotation in- 
  variant interactions. We will discuss two possible routes 
  to reduce this artiﬁcialness through controlled perturba- 
  tive expansions, by coupling to optical phonons or by 
  magnetic couplings between the elementary units. 