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

Density of states for a semi-infinite chain.

At the band center the DOS is constant: \(\rho_0=1/\pi t\). There is an alternative expression for the chain DOS by defining \(\cos\phi=\frac{\omega-\epsilon_0}{2t}\). Then \[g^r_{ch}(\omega)=\frac{1}{t}\left(\cos\phi-i\sin\phi\right)=\exp{(-i\phi)}{t}\] In general we get \[G^{r}_{n1}=\exp{-i\phi}{t}\]

Impurity coupled to the semi-infinite chain

Now we consider that the \(0\) state is coupled to the semi-infinite chain with \(V\) as the tunneling amplitude. The Dyson equation for the impurity site Green function reads \[G_{00}=g_d+g_{11}+V_{10}G_{10},\] The bare dot Green function is \[g_{d}=\frac{1}{\omega-\epsilon_d}\] Now we take \(V_{10}=V\). Then, the chain Green function \(G_{10}\) coupled to the impurity reads \[G_{10}=g_{10}+g_{10}V_{01}G_{00}=g_{ch}VG_{10},\] With \(g_{10}=0\) and \(V_{01}=V_{10}=V\). Replacing back \(G_{10}\) in the Dyson equation for \(G_{00}\) we have \[G_{00}=\frac{1}{g_d^{-1}-|V|^2 g_{ch}}\] For energies around the band center we can neglect the energy dependence of \(g_{ch}\approx \frac{-i}{t}\), then