Microscopic theory of superconductivity in the language of the vertex function In Chapter 7 of Abrikosov, Gorkov, and Dzyaloshinski (AGD), the authors cover a microscopic overview of superconductivity, with an emphasis on the poles of the vertex function $\Gamma$. Despite the thoroughness of their derivation, I would prefer to look at other such references of similar quality, preferably in a more modern context. Therefore, I am specifically looking for additional sources (original papers, review articles, books, etc.) that cover a microscopic approach to the Cooper instability, with an emphasis on its connection to the vertex function, the phonon propagator, and the corresponding diagrammatics. The more advanced the better, but I would prefer references with excessive detail if possible. Basically, a modern supplement to AGD subsections 7.33, 7.34, and 7.35 at the advanced graduate/postdoc level.
EDIT: Clarified question.
 A: To see Cooper instability from the vertex function, you can check Altland & Simons book. May be it is not a postdoc level but it seems that topic is not so hard. Here I sketch the derivation.
Consider 4-fermion theory with contact attractive interaction $g$. Then consider temperature Green functions and non-crossing approximation. It means that vertex function is given by infinite ladder,

Performing summation of geometric series, one can find the following expresstion for vertex function,
$$\Gamma_q=g+\frac{gT}{V}\sum_{p}\Gamma_qG_{p+q}G_{-p}.$$
In this notation, $q=(i\omega,{\bf q})$. Performing summation over Matsubara frequencies, we find
$$\int_p\frac{n_F(\xi_p)-n_F(\xi_{p+q})}{i\omega_n-\xi_{p+q}-\xi_p}\approx \nu_F\ln\frac{\omega_D}{T},$$
which indicates to Cooper instability. Our approximation about contact interaction relates to the phonon propagator.
Electron-phonon interaction hamiltonian is
$$H_{\text{ph-el}}=\gamma\int d^3rn(r)\partial\cdot{\bf u}(r)=\gamma\sum_{q\lambda}\frac{iq_{\lambda}}{\sqrt{2m\omega_q}}n_q(a_{q\lambda}+a^{\dagger}_{-q\lambda}),$$
where $\lambda$ is polarization and $n_q$ is electron density. Full hamiltonian contains free electrons kinetic term & phonon kinetic term. It is possible to integrate out phonon fields and find effective interaction. It will be
$$-\frac{\gamma^2}{2m}\sum_q\frac{q^2n_qn_{-q}}{\omega_n^2+\omega_q^2},$$
where $\omega_n$ is Matsubara frequency and $\omega_q$ is phonon dispersion law. Going to real and low frequencies, we see that effective interaction is
$$-\frac{\gamma^2}{2m}\sum_{q}n_{q}n_{-q},$$
which is exactly contact 4-fermion interaction. This derivation also exists in Altland books, see problem for chapter 4. This approximation is the same as in standard model, where you replace $W$-boson propagator by $1/m_w^2$ due to its large mass.
What do you want:


*

*See connection with diagrams and phonon propagator

*Understand how Cooper instability can be derived from vertex function & diagrams


So, what can you find in Altland:


*

*For low frequencies, we can replace phonon propagator by constant, which gives 4-fermion contact attractive interaction

*Summing geometric series of ladder diagrams, we can obtain the vertex function and this vertex function has the pole which is non-analytic expression in coupling constant $g$. It means that ground state of initial theory is reconstructed.


If you want more, I assume that you can check Gorkov papers in JETP or references in Altland book.
