text: chirality interactions in cold atom optical lattices has 
  been proposed38.  system. And convert them to pseudo-spin notation in 
  the physical spin singlet sector. 
  
  Our model (8) is achieved at second order of the per- 
  turbation series. Higher order terms become trunca- 
  tion errors but may be controlled by small parameters 
  λx,y,z/Jcluster 
  
  Consider a general small distortion of the tetrahedron, 
  the spin Hamiltonian becomes 
  
  Jx,y,z| 
  ′ 
  Sℓ)2 + J  Hcluster, SL = (Jcluster/2)( 
  δrℓm(Sℓ · 
  Sm) 
  ∼ p| 
  X 
  ℓ  X 
  ℓ<m 
  
  (A1) 
  where δrℓm is the change of bond length between spins 
  ℓ and m, J ′ is the derivative of Jcluster with respect to 
  bond length. 
  
  There are six orthogonal distortion modes of the tetra- 
  hedron [TABLE 1.1 in Ref.35]. One of the modes A is the 
  trivial representation of the tetrahedral group Td; two E 
  modes form the two dimensional irreducible representa- 
  tion of Td; and three T2 modes form the three dimen- 
  sional irreducible representation. The E modes are also 
  illustrated in FIG. 3. 