text: ·  −  · 
  
  Now we can use TABLE I to convert the above cou- 
  It is easy to see that f A and 
  plings into pseudo-spin. 
  f T2 
  1,2,3 are all zero when converted to pseudo-spins, namely 
  projected to the physical spin singlet sector. But f E 
  1 = 
  (√3/2)τ x 
  2P34)/(4√3) = 
  (P14 +P23 +P24 +P13 
  − 
  and f E 
  P23)/4 = (√3/2)τ y. This 
  2 = (P24 + P13 
  − 
  has already been noted by Tchernyshyov et al.28, only 
  the E modes can lift the degeneracy of the physical spin 
  singlet ground states of the tetrahedron. Therefore the 
  general spin lattice coupling is the form of (12) given in 
  the main text. 
  
  2P12 
  P14 
  − 
  − 
  
  − 