text: 3 
  
  Jcluster limit. So only the singlet sector remains in low 
  energy. 
  
  The singlet sector is then treated as a pseudo-spin-1/2 
  Hilbert space. From now on we denote the pseudo-spin- 
  1/2 operators as T = (1/2)~τ , with ~τ the Pauli matri- 
  ces. It is convenient to choose the following basis of the 
  pseudo-spin 
  
  TABLE I: Correspondence between physical spin operators 
  and pseudo-spin operators in the physical spin singlet sector of 
  the four antiferromagnetically coupled physical spins. Pjk = 
  2Sj 
  Sℓ) 
  are spin-chirality operators. Note that several physical spin 
  operators may correspond to the same pseudo-spin operator. 
  
  τ z =  1  = 
  |  ±  i 
  
  | ↑↓↓↑i(cid:17) 
  (3) 
  | ↑↑↓↓i  | ↑↓↑↓i 
  
  where ω = e2πi/3 is the complex cubic root of unity, 
  and other states on the right-hand-side(RHS) are 
  | ↓↓↑↑i 
  basis states of the four-spin system, in terms of Sz quan- 
  tum numbers of physical spins 1, . . . , 4 in sequential or- 
  der. This pseudo-spin representation has been used by 
  Harris et al. to study magnetic ordering in pyrochlore 
  antiferromagnets21. 