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I am reading “A Short Course on Topological Insulators” by János K. Asbóth. et.all., and want to calculate the Bulk and edge state of the SSH model (Chapter 1) to drive the energy spectrum in Fig. 1.4.

The Hamiltonian of the 1D SSH model is as follows: $$H=v\sum^N_{m=1}(|m,B \rangle \langle m,A|+h.c.)+w\sum^{N-1}_{m=1}(|m+1,A \rangle \langle m,B|+h.c.),$$ where there are N unit cells, each with two internal degrees of Freedom, i.e. $|A\rangle$ and $|B\rangle$. Then, suppose $w=1$ and $N=10$.
In the trivial state where $v \gt w$, I calculate the Bulk state using Born-Von-Karman boundary conditions and the following Bulk momentum space Hamiltonian: $$H(k)=\begin{pmatrix}0&v+we^{-ik}\\v+we^{ik}&0\end{pmatrix}$$ where gives $E(k)=\pm \sqrt{v^2+1+2v \cos k}$, for $k=\frac{2\pi}{N}, \frac{4\pi}{N}, …,2\pi (\text{with}\; N=10)$.

However, these relations would not reproduce Fig. 1.4. For example, in the special case of $v=w=1$ we expect the gapless spectrum while there is a gap in the figure. Moreover, since $\cos k$ gives some identical values for $k=\frac{2\pi}{N}, \frac{4\pi}{N},\ldots,2\pi$, some of the bands are overlapped and so, it doesn’t appear $20$ bands. Also, I don’t know how to calculate the edge sate for the topological case with $v \lt w$.

Could anyone please guide me where is my mistake?

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You are calculating in momentum space, which implicitly means using periodic boundary conditions or a chain that extends infinitely in both directions. There are no boundary states expected here as there is no boundary. The boundary states appear only if you have a boundary. (Physicsts say edge even though in 1D there are ends and not edges. States bound to the boundary is more correct.) You need to also look at open boundary conditions.

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  • $\begingroup$ Thank you Terry for the answer. The real Hamiltonian (please see the first equation) has boundaries. However, to calculate the Bulk states in the trivial case, I used the periodic boundary conditions and so the momentum space Hamiltonian (second equation). In fact, I have three questions; 1) Is my analysis, for the bulk states in the trivial case, correct ? 2) If it is correct, why does it differ with the result in the book. 3) How to calculate edge states in the topological case? $\endgroup$
    – H. Khani
    Jun 26, 2021 at 22:44
  • $\begingroup$ The book is plotting using open boundary conditions in a finite model and your calculation (looks correct) is for the infinite model. $\endgroup$ Jun 27, 2021 at 16:43
  • $\begingroup$ You are right. Thank you very much. $\endgroup$
    – H. Khani
    Jun 27, 2021 at 22:54
  • $\begingroup$ You can compute numerically the spectrum of the Hamiltonian with open boundary conditions. See (1.2) in that text. $\endgroup$ Jun 28, 2021 at 0:46

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