# Fourier transform from lattice site into $k$-space in Hubbard-Holstein model

Say I have a one-dimensional lattice with lattice constant $$a$$. With next nearest neighbor hopping (NNN) included, the hopping term that describes such system would be

$$H_{hop} = -t\sum_j(\hat c_{j+1}^{\dagger}\hat c_j + H.c.) -t_2\sum_j(\hat c_{j+2}^{\dagger}\hat c_j +H.c.)$$

When we Fourier transform from lattice site into $$k$$-space, according to the equation

$$\hat c_j= \sum_k \hat c_k\cdot e^{-i\vec k\cdot\vec R_j}$$ ,

we get

$$H_{hop}$$ = $$-\sum_k\hat c_{k}^{\dagger}\hat c_k(2t \cos(ka) -2t_2\cos(2ka))$$ .

This is very simple for 1D lattice. But what if I want to do it in a 3D cubic lattice? Since electron can be hopping in three dimension, can I just sum them up and write,

for NN hoppping $$\rightarrow -2t[\cos(k_xa)+\cos(k_ya)+\cos(k_za)]$$,

for NNN hopping $$\rightarrow -2t_2[\cos(2k_xa)+\cos(2k_ya)+\cos(2k_za)]$$,

It is very straight forward, but I find a different result involving terms like $$\cos(k_xa)\cos(k_ya)$$ in Table 1.. Am I missing something here?

If you are at the origin, your nearest neighbors are at $$(\pm a,0,0)$$, $$(0,\pm a,0)$$, and $$(0,0,\pm a)$$. You are essentially claiming that your next nearest neighbors are at $$(\pm 2a,0,0)$$, $$(0,\pm 2a,0)$$, and $$(0,0,\pm 2a)$$. Are you sure you don't have any other neighbors nearer than that?