text: In (16), we have been able to reduce the four spin in- 
  teractions in (8) to inter-cluster Heisenberg interactions, 
  and the six-spin interactions in (8) to inter-cluster spin- 
  chirality interactions. The inter-cluster Heisenberg cou- 
  plings in Hperturbation x,y may be easier to arrange. The  inter-cluster spin-chirality coupling in Hperturbation z ex- 
  plicitly breaks time reversal symmetry and is probably 
  harder to implement in solid state systems. However 
  spin-chirality order may have important consequences 
  in frustrated magnets36,37, and a realization of spin- 