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Rosa edited untitled.tex
about 8 years ago
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Inserting the expressions for the self-energies we get
\begin{eqnarray}
N^>(\omega)&=&(e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi}
[G_{\beta\nu}^r(\omega+\epsilon) \Sigma_{0,\nu\gamma}^>(\omega+\epsilon) G_{\gamma\mu}^r(\epsilon) \Sigma_{0,\mu\beta}^{<,h}(\epsilon)] [G_{\beta\nu}^r(\epsilon) \Sigma_{0,\nu\gamma}^>(\epsilon) G_{\gamma\mu}^r(\epsilon+\omega) \Sigma_{0,\mu\beta}^{<,h}(\epsilon+\omega)]
\\ \nonumber
&&[G_{\beta\nu}^r(\omega+\epsilon) \Sigma_{0,\nu\gamma}^>(\omega+\epsilon) G_{\gamma\mu}^<(\epsilon) \Sigma_{0,\mu\beta}^{a,h}(\epsilon)] &&[G_{\beta\nu}^r(\epsilon) \Sigma_{0,\nu\gamma}^>(\epsilon) G_{\gamma\mu}^<(\epsilon+\omega) \Sigma_{0,\mu\beta}^{a,h}(\epsilon+\omega)]
\\ \nonumber
&&[G_{\beta\nu}^>(\omega+\epsilon) \Sigma_{0,\nu\gamma}^a(\omega+\epsilon) G_{\gamma\mu}^r(\epsilon)\Sigma_{0,\mu\beta}^{<,h}(\epsilon)] &&[G_{\beta\nu}^>(\epsilon) \Sigma_{0,\nu\gamma}^a(\epsilon) G_{\gamma\mu}^r(\omega+\epsilon)\Sigma_{0,\mu\beta}^{<,h}(\omega+\epsilon)]
\\ \nonumber
&&[G_{\beta\nu}^>(\omega+\epsilon) \Sigma_{0,\nu\gamma}^a(\omega+\epsilon)
G_{\gamma\mu}^<(\epsilon) \Sigma_{0,\mu\beta}^{a,h}(\epsilon)] G_{\gamma\mu}^<(\epsilon+\omega) \Sigma_{0,\mu\beta}^{a,h}(\epsilon+\omega)]
\end{eqnarray}
\begin{eqnarray}
&&N^>(\omega)=(4e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi} \times \Biggr\{
[G_{\beta\nu}^r(\omega+\epsilon) [G_{\beta\nu}^r(\epsilon) \Gamma_{\nu\gamma}
G_{\gamma\mu}^r(\epsilon) G_{\gamma\mu}^r(\epsilon+\omega) \Gamma_{\mu\beta}
(1-f_e(\omega+\epsilon) f_{h}(\epsilon)] (1-f_e(\epsilon) f_{h}(\epsilon+\omega)] \\ \nonumber
&&\sum_{\lambda\delta}[G_{\beta\nu}^r(\omega+\epsilon) &&\sum_{\lambda\delta}[G_{\beta\nu}^r(\epsilon) \Gamma_{\nu\gamma}
G_{\gamma\lambda}^a(\omega) \Gamma_{\lambda\delta}G^a_{\delta\mu}(\epsilon)[i\Gamma_{\mu\beta}](1-f_e(\omega+\epsilon)(f_{h}(\epsilon)+f_e(\epsilon))] G_{\gamma\lambda}^a(\epsilon) \Gamma_{\lambda\delta}G^a_{\delta\mu}(\epsilon+\omega)[i\Gamma_{\mu\beta}](1-f_e(\omega+\epsilon)(f_{h}(\epsilon)+f_e(\epsilon))] \\ \nonumber
&&\sum_{\lambda\delta} G_{\beta\lambda}^r(\omega+\epsilon)\Gamma_{\lambda\delta} G_{\delta\nu}^a(\omega+\epsilon)[i\Gamma_{\nu\gamma}] G_{\gamma\mu}^r(\omega)(1-f_{e}(\omega+\epsilon)+1-f_{h}(\omega+\epsilon))f_h(\epsilon)\\ \nonumber
&&\sum_{\lambda\delta\theta\tau}[G_{\beta\lambda}^r(\omega+\epsilon)\Gamma_{\lambda\delta} G_{\delta\nu}^a(\omega+\epsilon)[i\Gamma_{\nu\gamma}] G_{\gamma\theta}^r(\epsilon)\Gamma_{\theta\tau} G_{\tau\mu}^a(\epsilon)[i\Gamma_{\mu\beta}](1-f_{e}(\omega+\epsilon)+1-f_{h}(\omega+\epsilon))(f_{e}(\epsilon)+f_{h}(\epsilon))] \Biggr\}
\end{eqnarray}