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次へ: ? 上へ: surface Green function, orthogonal 戻る: normal

use transfer matrix

Rewrite eq.(16)

$\displaystyle G_{n0}$ $\textstyle =$ $\displaystyle t_0 G_{n-1,0} + \tilde{t}_{0} G_{n+1,0}$ (31)
$\displaystyle t_0$ $\textstyle =$ $\displaystyle (\omega -H_{00})^{-1} H_{01}^\dag $ (32)
$\displaystyle \tilde{t}_{0}$ $\textstyle =$ $\displaystyle (\omega -H_{00})^{-1} H_{01}$ (33)

Then the relation between the next nearest neighbor is

$\displaystyle G_{n0}$ $\textstyle =$ $\displaystyle t_0 (t_0 G_{n-2,0} + \tilde{t}_{0} G_{n,0} ) + \tilde{t}_{0} (t_0 G_{n,0} + \tilde{t}_{0} G_{n+2,0} )$ (34)
$\displaystyle G_{n0}$ $\textstyle =$ $\displaystyle (I- t_0 \tilde{t}_{0} - \tilde{t}_{0} t_0 )^{-1} ( t_0 t_0 G_{n-2,0} + \tilde{t}_{0} \tilde{t}_{0} G_{n+2,0} )$ (35)

rewrite it
$\displaystyle G_{n0}$ $\textstyle =$ $\displaystyle t_1 G_{n-2,0} + \tilde{t}_{1} G_{n+2,0}$ (36)
$\displaystyle t_1$ $\textstyle =$ $\displaystyle (I- t_0 \tilde{t}_{0} - \tilde{t}_{0} t_0 )^{-1} (t_0)^2$ (37)
$\displaystyle \tilde{t}_{1}$ $\textstyle =$ $\displaystyle (I- t_0 \tilde{t}_{0} - \tilde{t}_{0} t_0 )^{-1} (\tilde{t}_{0})^2$ (38)

generally

$\displaystyle G_{n0}$ $\textstyle =$ $\displaystyle t_i G_{n-2^i,0} + \tilde{t}_{i} G_{n+2^i,0}$ (39)
$\displaystyle t_i$ $\textstyle =$ $\displaystyle (I- t_{i-1} \tilde{t}_{i-1} - \tilde{t}_{i-1} t_{i-1} )^{-1} (t_{i-1})^2$ (40)
$\displaystyle \tilde{t}_{i}$ $\textstyle =$ $\displaystyle (I- t_{i-1} \tilde{t}_{i-1} - \tilde{t}_{i-1} t_{i-1} )^{-1} (\tilde{t}_{i-1})^2$ (41)

at eq.(39), set $n=2^i$

\begin{displaymath} G_{2^i,0} = t_i G_{0,0} + \tilde{t}_{i} G_{2^{i+1},0} \end{displaymath} (42)

use eq.(42) iteratively

$\displaystyle G_{10}$ $\textstyle =$ $\displaystyle t_0 G_{0,0} + \tilde{t}_{0} G_{2,0}$ (43)
    $\displaystyle ( G_{2,0}= t_1 G_{0,0} + \tilde{t}_{1} G_{4,0} )$ (44)
  $\textstyle =$ $\displaystyle (t_0 + \tilde{t}_{0} t_1 ) G_{00} + \tilde{t}_{0} \tilde{t}_{1} G_{4,0}$ (45)
    $\displaystyle ( G_{4,0}= t_2 G_{0,0} + \tilde{t}_{2} G_{8,0} )$ (46)
  $\textstyle =$ $\displaystyle (t_0 + \tilde{t}_{0} t_1 +\tilde{t}_{0} \tilde{t}_{1} t_2 ) G_{0,0} + \tilde{t}_{0} \tilde{t}_{1} \tilde{t}_{2} G_{8,0}$ (47)
  $\textstyle = \dots =$ $\displaystyle ( t_0 + \tilde{t}_{0} t_1 +\tilde{t}_{0} \tilde{t}_{1} t_2 + \dot... ... + \tilde{t}_{0} \tilde{t}_{1} \dots \tilde{t}_{i-1} \tilde{t}_{i} G_{2^{i+1}0}$ (48)
  $\textstyle =$ $\displaystyle T_{i} G_{00} + \tilde{t}_{0} \tilde{t}_{1} \dots \tilde{t}_{i-1} \tilde{t}_{i} G_{2^{i+1}0}$ (49)
$\displaystyle T_i$ $\textstyle =$ $\displaystyle t_0 + \tilde{t}_{0} t_1 +\tilde{t}_{0} \tilde{t}_{1} t_2 + \dots + \tilde{t}_{0} \tilde{t}_{1} \dots \tilde{t}_{i-1} t_{i}$ (50)

If $\vert \tilde{t}_{0} \tilde{t}_{1} \dots \tilde{t}_{i-1}\cdots \vert\rightarrow 0$ for $i\rightarrow \infty$, 2


\begin{displaymath} G_{10} = T G_{00} \end{displaymath} (51)

where $T=T_i\vert_{i\rightarrow\infty}$

from eq.(15) using $T$

$\displaystyle (\omega -H_{00} )G_{00} - H_{01} G_{10}$ $\textstyle =$ $\displaystyle I$ (52)
$\displaystyle G_{00}$ $\textstyle =$ $\displaystyle (\omega -H_{00} - H_{01} T )^{-1}$ (53)

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次へ: ? 上へ: surface Green function, orthogonal 戻る: normal
kino 平成18年4月17日