# Expanding the Hamiltonian Matrix for Unit cell to that for Supercell

Suppose there is a two dimensional unit cell, containing 3 atoms without spin orbit coupling effect. Therefore, the Hamiltonian matrix for this unit cell is a $$3\times3$$ matrix and the format of each element in this unit-cell Hamiltonian is $$t\times e^{ikd_{ij}}$$. $$t$$, $$k$$ and $$d$$ are hopping parameter between different atoms, $$k$$ point coordinate and the bonding vector between different atoms $$i$$ and $$j$$ respectively. For instance, the Hamiltonian matrix is written below.

$$\begin{matrix}atom1 & atom2 & atom3\end{matrix}\\ \begin{vmatrix} \epsilon_1 & t\times e^{ikd_{12}} & t\times e^{ikd_{13}}\\ t\times e^{-ikd_{21}} & \epsilon_2 & t\times e^{ikd_{23}}\\ t\times e^{-ikd_{31}} & t\times e^{-ikd_{32}} & \epsilon_3 \end{vmatrix}$$

Now, this unit cell is expanded to a $$5\times5$$ supercell. Would anyone please tell me how to expand the Hamiltonian matrix for this unit cell to that for the $$5\times5$$ supercell?

My own understanding is that I just need to copy the Hamiltonian for this unit and repeat it by 25 times. Every time I copy the Hamiltonian of unit cell, I just need to past it onto the diagonal part and leave the off-diagonal part zero. Finally, the Hamiltonian for this $$5\times5$$ supercell is just a $$75\times75$$ sparse matrix ($$3\times5\times5=75$$). The format of this supercell Hamiltonian is written below.

$$\begin{equation} \begin{matrix}unitcell1 & unitcell2 & . & . & . & unitcell25\end{matrix}\\ \begin{vmatrix} H_{1}\\ &H_{2}&&\mbox{0}\\ &&H_{3}\\ &\mbox{0}&&\dots\\ &&&&H_{25} \end{vmatrix} \end{equation}$$

I am not sure whether my understanding is correct or not. Is it just simple copy-and-paste of the Hamiltonian matrix the for unit cell by 25 times?

Would anyone please give me some suggestions or hints on this issue?

Thank you very much in advance.

• I do not get your question. Could you please be much more specific what you consider. What is the system and the key effect you are interested in? Even a single atom has many energy levels. Therefore, it can be described by a Hamiltonian with many dimensions. Dec 19, 2020 at 20:59
• @Semoi Thank you for your reply. I modified my question. Would you please give me some suggestions on this Hamiltonian expansion? Thank you very much again. Dec 21, 2020 at 5:18
• @Semoi I have already modified my question with more details. Would you please vote to re-open my question? I really need help on this question, which has puzzled me for long time. Thank you very much. Dec 21, 2020 at 17:19