text: Similarly considering the following perturbation on y- 
  
  links, λ Hperturbation = λ[Sj1 
  − 
  · 
  Sj4)]. Following similar procedures we get the second 
  order perturbation from this term 
  
  (Sk3  Sk4) + r Sk1  (Sj3 
  −  · 
  
  In this Appendix we derive the second order pertur- 
  bations of inter-cluster Heisenberg and spin-chirality in- 
  teractions. The results can then be used to construct 
  (16). 
  λ2 
  6Jcluster h 
  + 2r 
  
  9 + 9r2 
  8  − 
  
  Pjk[Sj1 
  (Sj3 
  
  Sj4)][Sk1 
  − 
  Sk4 + r2 Sj3 
  
  (Sk3  Sk4)] 
  Pjk 
  
  · 
  Pjk(Sk3 
  · 
  9 + 9r2 
  8 
  Pjk(Sk3 
  
  · 
  
  − 
  Pjki 
  (3/2)  Sj4) 
  − 
  
  · 
  j τ y 
  
  · 
  
  λ2 
  6Jcluster h 
  (3/2) 
  
  + 2r (3/4)τ y  = 
  k 
  − 
  
  Sk4 + r2 Sj3  Sj4) 
  Pjki 