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3.3.2 Second Order in $ U$

At lower temperature or for larger $ U$, higher-order terms in the vertex are important. The second order corrections to the irreducible vertex function in the $ \lambda $-approximation (i.e. zero external momentum transfer $ q=0$ and frequency $ \nu_n=0$) are shown in Fig. 3.6. The irreducible vertex is given in the spin channel by

Figure: Second order diagrams for the vertex functions $ \Gamma^2_{\uparrow\uparrow}$ - a.) and $ \Gamma^2_{\uparrow\downarrow}$ - b.) for an external momentum transfer $ q=0$.
\includegraphics[width=0.9\textwidth,clip, clip]{2ndorderU.eps}

...\omega_n+i\omega_{n'}-i\omega_{n''},K_1+K_1'-K_1'') \end{split}\end{displaymath} (3.23)

and in the charge channel

...ega_n-i\omega_{n'}+i\omega_{n''},K_1-K_1'+K_1''))~. \end{split}\end{displaymath} (3.24)

© Cyrill Slezak