In the book "Piers Coleman - Introduction to Many-Body Physics (2016, Cambridge University Press)" http://download.library1.org/main/1558000/6a62454463a644d8b5cfa7936cf355de/Piers%20Coleman%20-%20Introduction%20to%20Many-Body%20Physics-Cambridge%20University%20Press%20%282016%29.pdf in page 25 can any one tell me how the term of potential energy is coming?


When you are summing over all lattice points of a one-dimensional lattice with periodic boundary conditions, you can shift the summation index. For example,

$$\sum_j f_j = \sum_j f_{j+1} = \sum_j f_{j-1}$$

because all of these sum over all the $f$'s.

Begin by expanding the square,

$$\sum_j(\phi_j-\phi_{j+1})^2 = \sum_j(\phi_j^2 - 2\phi_j\phi_{j+1} + \phi_{j+1}^2).$$

Rewrite this as

$$\sum_j(\phi_j^2 + \phi_{j+1}^2 - \phi_j\phi_{j+1} - \phi_j\phi_{j+1}).$$

In the second and fourth terms, shift $j \to j-1$, giving

$$\sum_j(\phi_j^2 + \phi_j^2 - \phi_j\phi_{j+1} - \phi_{j-1}\phi_j).$$

This then factorizes as

$$\sum_j \phi_j(2\phi_j - \phi_{j+1} - \phi_{j-1}).$$


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