# Different ways to defining Fourier transform of a lattice model

I have a Kagome 2D lattice as shown in the figure below:

There are three atoms in one unit-cell A, B, and C (shown in circle). An interaction Hamiltonian can be $$H = H_1+H_2+H_3\equiv -t\sum_{\langle i,j \rangle} a_i^+b_j -t\sum_{\langle i,j \rangle} b_i^+c_j -t\sum_{\langle i,j \rangle} c_i^+a_j$$ here $$a^+, b^+$$, and $$c^+$$ represents particle creation operators at site A,B, and C respectiviely. $$\langle i,j \rangle$$ represents nearest neighbours. Fourier transfer to operators can be defined a $$a_i^+ \to \sum_k a_k^+ e^{-i \hat{k}\cdot\hat{r_i}}$$. The first term becomes: $$-t\sum_{\langle i,j \rangle} \sum_k \sum_{l} a_k^+ b_{l} e^{i(-\hat{k}\cdot\hat{r_i} + \hat{l}\cdot\hat{r_j})}$$

As it can be seen from the lattice, every site A has two B neighboring sites. I have seen in literature, people choose two ways to choose neighbors. I want to know the difference between the two approaches.

First way (FT1):

choose the nearest neighbor B of site A such that one B is within the same uni-cell, for example, if we choose A site labeled as "X" (in the above figure) as center then the one nearest neighbor B site is highlighted as "1". The other B neighbor is in a different unit-cell, highlighted as "2".

This way, we get $$r_j = \{r_i, r_i+r_{X2}\}$$, and hence $$-t\sum_{\langle i,j \rangle} \sum_k \sum_{l} a_k^+ b_{l} e^{i(-\hat{k}\cdot\hat{r_i} + \hat{l}\cdot\hat{r_j})} = -t\sum_k a_k^+b_k (1+e^{i\hat{k}\cdot\hat{r}_{X2}})$$ where $$r_{X2}$$ is vector connecting site X and 2.

Second way (FT2):

Now the nearest neighbors are simple vector connecting X and 1 ($$\hat{r}_{X1}$$), and vector connecting X and 2 ($$\hat{r}_{X2}$$).

This way, we get $$r_j = \{r_i+r_{X1}, r_i+r_{X2}\}$$, and hence $$-t\sum_{\langle i,j \rangle} \sum_k \sum_{l} a_k^+ b_{l} e^{i(-\hat{k}\cdot\hat{r_i} + \hat{l}\cdot\hat{r_j})} = -t\sum_k a_k^+b_k ( e^{i\hat{k}\cdot\hat{r}_{X1}} + e^{i\hat{k}\cdot\hat{r}_{X2}})$$

Comparison of band structures:

Bandstructure does not change, except that the period of FT1 is double than the FT2

I am confused about how these two different approaches differ only by period?

• There usually different ways to choose an elementary cell in a crystal, leading to somewhat different expansions. These are typically related via crystal symmetries. Mar 29, 2021 at 2:13