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\textit{Oh, an empty article!}   You can get started by \textbf{double clicking} In  this text block and begin editing. You can also click the \textbf{Insert} button below note we derive the displacement charge of a site which is coupled  to add new block elements. Or you can \textbf{drag and drop an image} right onto a noninteracting tight binding semi-infinite chain.   We start with the following wavefunction  \begin{equation}  |\Psi\rangle =\sum_{\beta=1}^{\infty} \sin(k\beta)|\beta\rangle  \end{equation}  When The Hamiltonian of the chain $H$ acts on  this text. Happy writing! wavefunction, then, we get the eigenenergies  \begin{equation}  H\Psi\rangle = (\epsilon_0 + 2 t \cos k)|\Psi\rangle  \end{equation}  Here $k=2\pi/a$ with $a$ being the lattice spacing. Hereafter, we set $\epsilon_0=0$. Now we compute the Dyson equation for this semi-infinite chain. We do not consider now the site $0$.   \begin{equation}  G_{11}=g_{11}+g_{11} t_{12} G_{21}  \end{equation}  We consider all hopping amplitude for all sites the same, $t_{12}=t$. Here, $g_{11}$ is the bare Green function for an uncoupled site, ie.,   \begin{equation}  g_{11}=\frac{1}{\omega-\epsilon_0}  \end{equation}