If site0 and site1 are in the central region

Matrixes for site0 and site1 are

$$H = \left( \begin{array}{cc} 0&t t&0 \end{array} \right)$$ (239)

$$\Sigma_L(\omega ^-) = \left( \begin{array}{cc} \omega _0/2 + i\Gamma_0/2 & 0 \\ 0&0 \end{array} \right)$$ (240)

$$\Sigma_R(\omega ^-) = \left( \begin{array}{cc} 0&0 0& \omega _0/2 + i\Gamma_0/2 \end{array} \right)$$ (241)

$$\Gamma _L = \left( \begin{array}{cc} \Gamma_0 & 0 \\ 0 & 0 \end{array} \right)$$ (242)

$$\Gamma _R = \left( \begin{array}{cc} 0 & 0 \\ 0 & \Gamma_0 \end{array} \right)$$ (243)

$$G_C(\omega ^+) = \left( \omega - H - \Sigma_R(\omega ^+) - \Sigma_L(\omega ^+) \right)^{-1}$$ (244)

Then

$$T = {\rm Tr}[ \Gamma_L G_C(\omega ^+) \Gamma_R G_C(\omega ^-) ]$$ (245)

$$= \frac{16 \Gamma _0^2 t^2 }{ ( \Gamma _0^2 + (2t +\omega )^2) (\Gamma _0^2+(2t-\omega )^2) }$$ (246)

where $\omega _0=\omega$ is used. $T=1$ also in this case. Then

$$\Gamma _0 = \sqrt{ 4 t^2 - \omega ^2 }$$ (247)

A numerical calculation supprots this result.

In summary, self-energy of the surface Green function of the simple tight-binding model is

$$\Sigma_{R { \rm or } L}(\omega ^\pm)= \frac{\omega }{2} \mp i \frac{\sqrt{4 t^2-\omega ^2}}{2}$$ (248)

for $-2t <\omega <2t$.

図 2: Self-energy of surface Green function of the simple tight binding model of eq.(232) at $t=1$. A figure shows a result of numerical calculation of $\Sigma (\omega +i\delta )$ where $\delta =0.01$.