I have a Hamiltonian in the momentum space which has a strong non-local electron-electron interaction.
I know that I have to find its corresponding exchange self-energy and solve the Dyson equation self-consistently.
$$G = G_0 * \Sigma_I * G$$
which $\Sigma$ is a function of $G$. $\quad$ ($\Sigma(G)$)
So how does this self-energy look like in the momentum space. If there is not a specific self-energy, how should I start to make it step by step.