text: where λ (in fact λ/k) is the expansion parameter. 
  Consider the perturbation Hperturbation = QE 
  
  QE 
  1k, 
  1j · 
  which means a coupling between the QE 
  1 distortion 
  modes of the two tetrahedra. Integrate out the optical 
  phonons, at lowest non-trivial order, it produces a term 
  (3 J ′2 λ)/(4 k2) τ x 
  τ x 
  k . This can be seen by minimizing 
  j · 
  separately the two cluster Hamiltonians with respect to 
  1 = (√3 J ′)/(2 k)τ x, then plug this 
  QE 
  into the perturbation term. Thus we have produced the 
  (3 J ′2 λ)/(4 k2). 
  Jx term in the Kitaev model with Jx = 
  QE 
  2k 
  τ y 
  k at lowest non-trivial 
  
  1 , which gives QE 
  
  − 
  Similarly the perturbation Hperturbation = QE 
  2j · 
  
  will generate (3 J ′2 λ)/(4 k2) τ y 
  j · 
  (3 J ′2 λ)/(4 k2). 
  order. So we can make Jy = 
  − 
  τ z 
  k coupling is more diﬃcult to get. We 
  j τ y 
  τ x 
  k . By the above reasoning, we 
  j · 
  2j · 
  It will produce at lowest non-trivial or- 
  k . Thus we have Jz = 
  j τ y 
  j · 
  k τ y 
  τ x 