text: of the two clusters can generate at lowest order the de- 
  sired high order spin interactions. In Subsection IV B we 
  will introduce certain magnetic, e.g. Heisenberg-type, in- 
  teractions between physical spins of diﬀerent clusters, at 
  lowest order(second order) of perturbation theory the de- 
  sired high order spin interactions can be achieved. These 
  approaches involve truncation errors in the perturbation 
  series, thus the mapping to low energy eﬀect Hamilto- 
  nian will no longer be exact. However the error intro- 
  duced may be controlled by small expansion parameters. 
  In this Section we denote the physical spins on cluster 
  j(k) as j1, . . . , j4 (k1, . . . , k4), and denote pseudo-spins 
  on cluster j(k) as ~τj (~τk). 
  
  It is tempting to call this as an exactly solved spin liq- 
  uid with spin gap ( 
  Jcluster), an extremely short-range 
  resonating valence bond(RVB) state, from a model with 
  spin rotation and time reversal symmetry. However it 
  should be noted that the unit cell of this model contains 
  an even number of spin-1/2 moments (so does the orig- 
  inal Kitaev model) which does not satisfy the stringent 
  deﬁnition of spin liquid requiring odd number of elec- 
  trons per unit cell. Several parent Hamiltonians of spin 
  liquids have already been constructed. See for example, 
  Ref.24–27. 
  
  ∼ 