Dyson equation self-consistent solution

These are given in Eqs. (C29) and (C38). The self-energy expressions (C38) and (C40) are calculated perturbatively to second order in the electron-phonon coupling in terms of the zeroth order Green functions (Eq. (55)). The simplest expression for current is obtained by substituting Eqs. (55), (C29) and (C38) in Eq. (51). This zeroth order result can be improved by using the renormalized Green functions obtained from the self-consistent solution of the Dyson equation (44). 

The procedure for determining the self-consistent solution of the Dyson equation can now be summarized as follows ... 

In the diagonal, second-order approximation to the self-energy of the electron propagator, solutions of the Dyson equation (with self-consistent pole energies, cvp) satisfy... 

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