How Does This Quantum Spin Equation Simplify?

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\frac{1}{2}S\sum_{\vec{n},\vec{m}}I_{\vec{n}-\vec{m}}[(S-S_{\vec{n}}^z)+(S-S_{\vec{m}}^z)]=SI(0)\sum_{\vec{m}}(S-S_{\vec{m}}^z)

How can I get this result?
 
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Separate the terms into two different sums, then you will have one sum that looks like \sum_{n,m} I_{n-m} (S-S^z_n) and another with S^z_m. For the former, the sum over m can be carried out.
 
Petar Mali said:
\frac{1}{2}S\sum_{\vec{n},\vec{m}}I_{\vec{n}-\vec{m}}[(S-S_{\vec{n}}^z)+(S-S_{\vec{m}}^z)]=SI(0)\sum_{\vec{m}}(S-S_{\vec{m}}^z)

What does this equation describe (just curious)?
 

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