Can I ask a question about the Hamiltonian expansion.

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.

  • $\begingroup$ 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. $\endgroup$
    – Semoi
    Dec 19, 2020 at 20:59
  • $\begingroup$ @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. $\endgroup$
    – Kieran
    Dec 21, 2020 at 5:18
  • $\begingroup$ @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. $\endgroup$
    – Kieran
    Dec 21, 2020 at 17:19


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