text: S1 + S2 + S3 + S4)2  (11) 
  
  · 
  with Jcluster > 0 and 0 < r < 3. However this is not 
  convenient for later discussions and will not be used. 
  
  We brieﬂy describe some of the properties of (8). Its 
  low energy states are entirely in the space that each of the 
  clusters is a physical spin singlet (called cluster singlet 
  subspace hereafter). Therefore physical spin correlations 
  are strictly conﬁned within each cluster. The excitations 
  carrying physical spin are gapped, and their dynamics 
  are ‘trivial’ in the sense that they do not move from one 
  cluster to another. But there are non-trivial low energy 
  physical spin singlet excitations, described by the pseudo- 
  spins deﬁned above. The correlations of the pseudo-spins 
  can be mapped to correlations of their corresponding 
  physical spin observables (the inverse mappings are not 
  unique, c.f. TABLE I). For example τ x,y correlations 
  become certain dimer-dimer correlations, τ z correlation 
  becomes chirality-chirality correlation, or four-dimer cor- 
  relation. It will be interesting to see the corresponding 
  picture of the exotic excitations in the Kitaev model, e.g. 
  the Majorana fermion and the Ising vortex. However this 
  will be deferred to future studies. 