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I would like to understand why we have the following equality when doing sommation on nearest neighbour :

$$ \sum_{<i,j>}=\frac{1}{2} \sum_i \sum_{j=ppv(i)}$$

$ <i,j>$ represents summation on the pairs and $ ppv(i) $ the nearest neighbour of $i$.

I see in a 1D lattice why it is true but I don't understand why it is true in a general case.

[edit] In fact I think I get it... To sum on pairs is like summing on everything but we have to divide by $2$ because if $i_0$ is near from $j_0$, then $j_0$ is near from $i_0$ and as we sum on all $i$ we will be on $i_0$ at a time and on $j_0$ at another time (so we count twice a term), and as we only look for nearest neighbours pair we have the $ppv(i)$, right ?

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    $\begingroup$ Yes, the half comes from the fact that the sums are over each bond twice. $\endgroup$ – lemon Nov 9 '16 at 11:06
  • $\begingroup$ It would be good if you answer your own question (=post this as an answer). $\endgroup$ – Norbert Schuch Nov 9 '16 at 12:23
  • $\begingroup$ Okay I'll do this :) $\endgroup$ – StarBucK Nov 9 '16 at 12:40
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In fact I think I get it...

To sum on pairs is like summing on everything but we have to divide by 2 because if $i0$ is near from $j0$, then $j0$ is near from $i0$ and as we sum on all $i$ we will be on $i0$ at a time and on $j0$ at another time (so we count twice a term), and as we only look for nearest neighbours pair we have the $ppv(i)$.

In conclusion, the formula is valid in any dimension and with any number of nearest neighbours.

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