text: Another note to take is that it is not necessary to have 
  such a highly symmetric cluster Hamiltonian (2). The 
  mappings to pseudo-spin-1/2 should work as long as the 
  ground states of the cluster Hamiltonian are the two-fold 
  degenerate singlets. One generalization, which conforms 
  the symmetry of the lattice in FIG. 2, is to have 
  
  Hcluster = (Jcluster/2)(r 