text: In this Subsection we regard each four-spin cluster 
  as a tetrahedron, and consider possible optical phonon 
  modes(distortions) and their couplings to the spin sys- 
  tem. The basic idea is that the intra-cluster Heisen- 
  berg coupling Jcluster can linearly depend on the dis- 
  tance between physical spins. Therefore certain distor- 
  tions of the tetrahedron couple to certain linear combi- 
  Sm. Integrating out phonon modes will 
  nations of Sℓ · 
  then generate high order spin interactions. This idea has 
  been extensively studied and applied to several magnetic 
  materials28–34. More details can be found in a recent 
  review by Tchernyshyov and Chern35. And we will fre- 
  quently use their notations. In this Subsection we will 
  use the representation (5) for τ z. 
  
  IV. GENERATE THE HIGH ORDER PHYSICAL 
  SPIN INTERACTIONS BY PERTURBATIVE 
  EXPANSION. 
  
  One major drawback of the present construction is that 
  it involves high order interactions of physical spins[see 
  (8) and (9)], thus is ‘unnatural’. In this Section we will 
  make compromises between exact solvability and natu- 
  ralness. We consider two clusters j and k and try to 
  generate the Jx,y,z interactions in (7) from perturbation 
  series expansion of more natural(lower order) physical 
  spin interactions. Two diﬀerent approaches for this pur- 
  pose will be laid out in the following two Subsections. In 
  Subsection IV A we will consider the two clusters as two 
  tetrahedra, and couple the spin system to certain opti- 
  cal phonons, further coupling between the phonon modes 