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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?

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

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