text: λ2 
  
  2Jcluster Pjk(Hperturbation)2 
  This is true for other perturbations considered later in 
  this Appendix. The cluster j and cluster k parts can be 
  separated, this term then becomes (a, b = x, y, z), 
  
  Pjk  − 
  
  =λy[Sj1 
  Jy| 
  Jcluster, r = sgn(Jy) is the sign of Jy. 
  4 
  with λy = 
  p 
  j τ z 
  The τ z 
  k term is again more diﬃcult to get. We use 
  the representation of τ z by spin-chirality (6). And con- 
  sider the following perturbation 
  
  (cid:2)PjSa 
  2Jcluster X 
  a,b 
  j2Pj · PkSa 
  j1Sb 
  PjSa 
  j2Pj · PkSa 
  j2Sb 
  PjSa 
  jℓSb 
  PjSa 
  Pj(Sjℓ · 
  Then use the fact that 
  Sjm) 
  Pj by spin rotation symmetry, the perturbation be- 
  comes 
  
  j1Pj · PkSa 
  k1Sb 
  k2Sb 
  
  k1Pk  − 
  
  Hperturbation = Sj2  (Sj3  Sj4) + r Sk2  (Sj3  Sj4) 
  ·  ·  × 
  
  + 2r 
  + r2 
  
  k2Pk 
  k2Pk(cid:3) 
  jmPj = δab(1/3) 
  λ2 
  · 