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 $q=0$.

$$\begin{split}\Gamma^{(\lambda)s}(i\omega_n,i\omega_{n'};K_1&,... ...\omega_n+i\omega_{n'}-i\omega_{n''},K_1+K_1'-K_1'') \end{split}$$ (3.23)

and in the charge channel

$$\begin{split}\Gamma^{(\lambda)c}(i\omega_n,i\omega_{n'}&;K_1,... ...ega_n-i\omega_{n'}+i\omega_{n''},K_1-K_1'+K_1''))~. \end{split}$$ (3.24)