text: modes of neighboring tetrahedra. And these coupling 
  constants λx,y,z need to be tuned to produce Jx,y,z of 
  the Kitaev model. This is still not easy to implement in 
  solid state systems. At lowest non-trivial order of pertur- 
  bative expansion, we do get our model (9). Higher order 
  terms in expansion destroy the exact solvability, but may 
  be controlled by the small parameters λx,y,z/k. 
  
  the desired Jx,y,z terms in (8) from the ﬁrst and second 
  order of perturbations. 
  
  The calculation can be dramatically simpliﬁed by the 
  following fact that any physical spin-1/2 operator Sx,y,z 
  ℓ 
  converts the cluster spin singlet states 
  into 
  spin-1 states of the cluster. This can be checked by 
  explicit calculations and will not be proved here. For 
  all the perturbations to be considered later, the above 
  mentioned fact can be exploited to replace the factor 
  Hcluster k]−1 in the second order pertur- 
  [0 
  Hcluster j − 
  bation to a c-number ( 
  
  τ z =  1 
  |  ±  i 
  
  2Jcluster)−1. 
  The detailed calculations are given in Appendix B. We 
  − 