text: derived as well. There have been several proposals to 
  open the fermion gap for the non-Abelian phase without 
  spoiling exact solvability4,6. And many generalizations 
  to other(even 3D) lattices have been developed in the 
  last few years10–16. All these eﬀorts have signiﬁcantly 
  enriched our knowledge of exactly solvable models and 
  quantum phases of matter. 
  
  FIG. 2: Left: the physical spin lattice for the model (8). The 
  dash circles are honeycomb lattice sites, each of which is ac- 
  tually a cluster of four physical spins. The dash straight lines 
  are honeycomb lattice bonds, with their type x, y, z labeled. 
  The interaction between clusters connected by x, y, z bonds 
  are the Jx,y,z terms in (8) or (9) respectively. Note this is not 
  the 3-12 lattice used in Ref.9,10. Right: enlarged picture of 
  the clusters with the four physical spins labeled as 1, . . . , 4. 
  Thick solid bonds within one cluster have large antiferromag- 
  netic Heisenberg coupling Jcluster. 