text: First consider the perturbation λ Hperturbation = λ[Sj1 
  · 
  Sk1 + r(Sj2 
  Sk2)], where r is a real number to be tuned 
  later. Due to the fact mentioned in Subsection IV B, 
  the action of Hperturbation on any cluster singlet state 
  will produce a state with total spin-1 for both cluster j 
  and k. Thus the ﬁrst order perturbation in (15) van- 
  ishes. And the second order perturbation term can be 
  Hcluster j − 
  − Pjk)[0 
  greatly simpliﬁed: operator (1 
  Hcluster k]−1(1 
  − Pjk) can be replaced by a c-number 
  2Jcluster)−1. Therefore the perturbation up to second 
  ( 
  − 
  order is 
  
  − 
  So we can choose 
  the last intra-cluster Sk3 
  ﬁrst order perturbation. 
  · 
  (r λ2)/(4Jcluster) = 
  Sk4 + r2 Sj3 
  
  · 
  
  Jy, and include 
  Sj4 term in the 
  − 
  − 
  ·  · 
  
  Therefore we can choose the following perturbation on 
  y-links (not unique), 