# Calculation of the displacement charge in a semi-infinite chain coupled to a localized impurity

In this note we derive the displacement charge of a site which is coupled to a noninteracting tight binding semi-infinite chain. First we derive the Green function for a semi-infinite chain and after we coupled such chain to an impurity with a energy level $$\epsilon_d$$. $H=\sum_{\sigma} \epsilon_0 c^\dagger_{i} c_{i} + t \sum_{n=1}^{n=\infty} \left(c^\dagger_{i}c_{i+1}+h.c\right)$

# Semi-infinite chain Green function

We start with the following wavefunction $|\Psi\rangle =\sum_{\beta=1}^{\infty} \sin(k\beta)|\beta\rangle$ When, he Hamiltonian of the chain $$H$$ acts on this wavefunction, then, we get the eigenenergies $H\Psi\rangle = (\epsilon_0 + 2 t \cos k)|\Psi\rangle$ 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 $$0$$. $G_{11}=g_{11}+g_{11} t_{12} G_{21}$ 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., $g_{11}=\frac{1}{\omega-\epsilon_0}$ Under these considerations we have $G_{21}=g_{21} + g_{22} t_{21} G_{11} + g_{22} t_{23} G_{31},$ and similar equations for $$G_{31}$$, etc. This sequence of Green functions for $$G_{n1}$$ generates an infinite number of equations. Now, it comes the trick. Consider that site $$2$$ is in reality another semi-infinite chain coupled to site $$1$$. Then, the equation for $$G_{21}$$ becomes simply $G_{12}=g_{ch} t_{21} G_{11}.$ By replacing this expression for $$G_{12}$$ in $$G_{11}$$ we can close the Dyson equation and to obtain explicitly $$G_{11}$$. Therefore $G_{11}=g_{11}+ g_{11}|t|^2 g_{ch} G_{11}.$ Besides, we notice that $$G_{11}$$ corresponds to a semi-infinite chain itself, thus $g_{ch}=g_{11}+g_{11}|t|^2 g_{ch}^2$ With the notorious result $g_{ch}=\frac{\omega-\epsilon_0}{2t^2}\pm\frac{1}{2t^2} \sqrt{(\omega-\epsilon_0)^2-4t^2},$ Considering $$|\omega-\epsilon_0|<2t$$, then $$g_{ch}$$ acquires an imaginary part, then $g_{ch}=\frac{\omega-\epsilon_0}{2t^2}\pm\frac{i}{t} \sqrt{1-\left(\frac{\omega-\epsilon_0}{2t}\right)^2},$ With a DOS $\rho_{ch}(\omega)=-\frac{1}{\pi}= \frac{1}{\pi t} \sqrt{1-\left(\frac{\omega-\epsilon_0}{2t}\right)^2},$ This is the surface (boundary) DOS, which never diverges in contrast with the bulk DOS