text: k 
  2 
  J ′(QE 
  
  k 
  2 
  1j)2 + 
  (QE  Hcluster j, SL =Hcluster j + 
  
  √3 
  2 
  2jτ y  1jτ x 
  QE 
  j ), 
  j −  − 
  
  where k > 0 is the elastic constant for these phonon 
  modes, J ′ is the spin-lattice coupling constant, QE 
  1j and 
  QE 
  1 and QE 
  2j are the generalized coordinates of the QE 
  2 
  distortion modes of cluster j, Hcluster j is (2). As al- 
  ready noted in Ref.35, this model does not really break 
  the pseudo-spin rotation symmetry of a single cluster. 
  
  where QE 
  1 mode 
  on cluster j, and QE 
  2k are similarly deﬁned; 
  (4Jx,yk2)/(3J ′2) and λz = (16Jzk4)/(9J ′4); the 
  λx,y = 
  single cluster spin-lattice Hamiltonian Hcluster, SL is (13). 
  Collect the results above we have the spin-lattice 
  
  Now we put two clusters j and k together, and in- 
  clude a perturbation λ Hperturbation to the optical phonon 
  Hamiltonian, 
  
  Hjk,SL =Hcluster j, SL + Hcluster k, SL 
  + λ Hperturbation[QE 
  1j, QE  2j, QE  1k, QE 
  2k] 