text: × 
  The ﬁrst order term in (15) vanishes due to the same 
  reason as before. There are four terms in the second 
  order perturbation. The ﬁrst one is 
  PjkSj2 
  [0 
  − 
  − Pjk)Sj2 
  (1 
  
  (Sk3 
  × 
  Hcluster j − 
  Sk4)(1 
  − Pjk) 
  Hcluster k]−1 
  Sk4) 
  × 
  × 
  (Sk3 
  Pjk 
  
  · 
  For the cluster j part we can use the same arguments 
  as before, the Hcluster j can be replaced by a c-number 
  Jcluster. For the cluster k part, consider the fact that 
  Sk4], 
  Sk3 
  the action of Sk3 
  Sk4 on physical singlet states of k will 
  also only produce spin-1 state. So we can replace the 
  Hcluster k in the denominator by a c-number Jcluster as 
  well. Use spin rotation symmetry to separate the j and 
  k parts, this term simpliﬁes to 
  
  × 
  Pjki  − 
  
  = 
  − 
  
  λ2 
  The perturbation on x-links is then (not unique), 
  Sj2 
  Pj · Pk(Sk3 
  Sk4)  (Sk3  Sk4) 
  Pk.  ·  ×  ·  × 
  Sk2)] 
  · 