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次へ: an 1D chain with 上へ: an 1D chain 戻る: If site 0 is

If site0 and site1 are in the central region

Matrixes for site0 and site1 are
$\displaystyle H$ $\textstyle =$ $\displaystyle \left( \begin{array}{cc} 0&t  t&0 \end{array} \right)$ (239)
$\displaystyle \Sigma_L(\omega ^-)$ $\textstyle =$ $\displaystyle \left( \begin{array}{cc} \omega _0/2 + i\Gamma_0/2 & 0 \\
0&0 \end{array} \right)$ (240)
$\displaystyle \Sigma_R(\omega ^-)$ $\textstyle =$ $\displaystyle \left( \begin{array}{cc} 0&0   0& \omega _0/2 + i\Gamma_0/2
\end{array} \right)$ (241)
$\displaystyle \Gamma _L$ $\textstyle =$ $\displaystyle \left( \begin{array}{cc} \Gamma_0 & 0 \\
0 & 0 \end{array} \right)$ (242)
$\displaystyle \Gamma _R$ $\textstyle =$ $\displaystyle \left( \begin{array}{cc} 0 & 0 \\
0 & \Gamma_0 \end{array} \right)$ (243)
$\displaystyle G_C(\omega ^+)$ $\textstyle =$ $\displaystyle \left( \omega - H - \Sigma_R(\omega ^+) - \Sigma_L(\omega ^+) \right)^{-1}$ (244)

Then
$\displaystyle T$ $\textstyle =$ $\displaystyle {\rm Tr}[ \Gamma_L G_C(\omega ^+) \Gamma_R G_C(\omega ^-) ]$ (245)
  $\textstyle =$ $\displaystyle \frac{16 \Gamma _0^2 t^2 }{ ( \Gamma _0^2 + (2t +\omega )^2) (\Gamma _0^2+(2t-\omega )^2) }$ (246)

where $\omega _0=\omega $ is used. $T=1$ also in this case. Then
\begin{displaymath}
\Gamma _0 = \sqrt{ 4 t^2 - \omega ^2 }
\end{displaymath} (247)

A numerical calculation supprots this result.

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

$\displaystyle \Sigma_{R {  \rm or  } L}(\omega ^\pm)= \frac{\omega }{2}
\mp i \frac{\sqrt{4 t^2-\omega ^2}}{2}$     (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 $\delta =0.01$.
\begin{figure}\begin{center}
\epsfig{file=Surf.eps,width=6cm}
\end{center}
\end{figure}


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次へ: an 1D chain with 上へ: an 1D chain 戻る: If site 0 is
kino 平成18年4月17日