Statistical mechanics, spin average

[K space]

This is an exercise from an old examination that I am trying to understand by following the given solution to this problem. The exercise is about to calculate the four spin average <S_iS_jS_kS_l> and the indices i,j,k and l, given a Hamiltonian H.

Solution:

The problem is that i can't follow the last two steps in this given solution. I understand how the partition function Z is obtained and how they express the four spin average as the trace of the four spins and the "density operator" (e^(-beta*H)/Z). But I am very confused about how they continue to the next last and last rows.

I'm thankful for explanations!

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?