Starting such an iterative solution, one can first neglect vertex corrections in (F.23) and obtain an approximation for the self-energy by means of (F.14) together with (F.21) and (F.22). Making use of this approximation, one calculates , and includes vertex corrections in the next step. The iteration of such a procedure generates an expansion in terms of the screened interaction and the GREEN's function defined as a self-consistent solution of the DYSON equation.
For the iterative procedure, the sequence of steps can be defined by the vertex function (F.23), which yields by means of the chain rule the recurrence relation
For the Self-consistent approximations, one selects a certain class of self-energy diagrams . The DYSON equation becomes a non-linear functional equation of the GREEN's functions, which has to be solved self-consistently. The selection corresponds to the summation of a certain class of diagrams up to infinite-order in the interaction, whereas others which contribute even in lower order are neglected. The difficulty is in finding the correct way to choose a subset of diagrams for each order. In order to deliver physically meaningful results, any approximation should guarantee certain macroscopic conservation laws. This condition can be imposed by the postulate that all diagrams contributing to the self-energy are obtained from the functional derivative of a functional with respect to . Solving the DYSON equation self-consistently with a -derivable self-energy yields a GREEN's function which conserves particle number, energy, and momentum .M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors