text: where τ x,y,z are Pauli matrices, and x, y, z-links are de- 
  ﬁned in FIG. 1. It was shown by Kitaev1 that this spin- 
  1/2 model can be mapped to a model with one Majo- 
  rana fermion per site coupled to Ising gauge ﬁelds on the 
  links. And as the Ising gauge ﬂux has no ﬂuctuation, the 
  model can be regarded as, under each gauge ﬂux conﬁg- 
  uration, a free Majorana fermion problem. The ground 
  state is achieved in the sector of zero gauge ﬂux through 
  each hexagon. The Majorana fermions in this sector have 
  Dirac-like gapless dispersion resembling that of graphene, 
  as long as 
  satisfy the triangular rela- 
  Jz| 
  tion, sum of any two of them is greater than the third 
  one1. It was further proposed by Kitaev1 that opening of 
  fermion gap by magnetic ﬁeld can give the Ising vortices 
  non-Abelian anyonic statistics, because the Ising vortex 
  will carry a zero-energy Majorana mode, although mag- 
  netic ﬁeld destroys the exact solvability. 
  
  Jx| 
  , 
  Jy| 
  , and 
  |  |  | 