Bethe-Salpeter Equation
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In Condensed Matter Physics, Optical Spectra of Insulators and Semiconductor are strongly affected by the actractive interaction between the photo-excited electron (in the conduction band) and the hole that it leaves behind in the valence band (Excitonic Effects). In some systems, like in solids of rare gases, the electron and the hole interact so strongly that they can form bound states located inside the gap (Excitons).
To describe electron-hole interaction (excitonic) effects, one should resort to a two-particle formalism, solving the Bethe Salpeter Equation (BSE) for the two-particle Correlation Function L. The results are very good. A short introduction to the Bethe-Salpeter Equation method can be found here.
\bgroup\color{red}$ \displaystyle \color{black} \color{red} L=GG + GG\Xi L $\egroup



\begin{displaymath} \Xi = \frac{\delta \Sigma}{\delta G} \simeq -i v + i W \end{displaymath}



The authorship and the responsability of the informations hereby provided belongs to Valerio Olevano.

Definitions

$L \quad$ 2-particle Correlation Function:

$ L(1,2,3,4) = - G^{(2)}(1,2,3,4) + G(1,3) G(2,4) $

$G \quad$ Green Function

$G^{(2)} \quad$ 2-particle Green Function


$P \quad$ Polarizability

$P(1,2) = -i L(1,2,1^+,2^+)$


$\Xi \quad$ Interaction Kernel


$\varepsilon \quad$ Dielectric Function

$\varepsilon(\omega) = 1 - v P(\omega)$

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