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\begin{equation}  g_{ch}=\frac{\omega-\epsilon_0}{2t^2}\pm\frac{1}{2t^2} \sqrt{(\omega-\epsilon_0)^2-4t^2},  \end{equation}  Considering $|\omega-\epsilon_0|<2t$, then $g_{ch}$ acquires an imaginary part, then  \begin{equation}  g_{ch}=\frac{\omega-\epsilon_0}{2t^2}\pm\frac{i}{t} \sqrt{1-\left(\frac{\omega-\epsilon_0}{2t}\right)^2},  \end{equation}  With a DOS  \begin{equation}  \rho_{ch}(\omega)=-\frac{1}{\pi}= \frac{1}{\pi t} \sqrt{1-\left(\frac{\omega-\epsilon_0}{2t}\right)^2},  \end{equation}  This is the surface (boundary) DOS, which never diverges in contrast with the bulk DOS